Tutorial
Release 10.4

The Sage Development TeamJul 23, 2024

CONTENTS

1

Introduction
1.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Ways to Use Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Longterm Goals for Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
4
5
5

2

A Guided Tour
2.1 Assignment, Equality, and Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Getting Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Functions, Indentation, and Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Basic Algebra and Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Some Common Issues with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Basic Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Parents, Conversion and Coercion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Finite Groups, Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Some More Advanced Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7
10
13
19
28
33
39
43
49
56
64
67
71

3

The Interactive Shell
87
3.1 Your Sage Session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Logging Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3 Paste Ignores Prompts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4 Timing Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 Other IPython tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.6 Errors and Exceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.7 Reverse Search and Tab Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.8 Integrated Help System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.9 Saving and Loading Individual Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.10 Saving and Loading Complete Sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4Interfaces
4.1 GP/PARI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 GAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Singular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Sage, LaTeX and Friends
115
5.1 Basic Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

107
107
109
110
112

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5.2
5.3
5.4

Customizing LaTeX Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Customizing LaTeX Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
SageTeX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6

Programming
6.1 Loading and Attaching Sage files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Creating Compiled Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Standalone Python/Sage Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Lists, Tuples, and Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Iterators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Loops, Functions, Control Statements, and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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123
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125
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131
133
134
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7

Using SageTeX
7.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Make SageTeX known to TeX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 SageTeX documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 SageTeX and TeXLive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Afterword
8.1 Why Python? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 I would like to contribute somehow. How can I? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 How do I reference Sage? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145
145
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9

Appendix149
9.1 Arithmetical binary operator precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

10 Bibliography151

11 Indices and tables

153

Bibliography

155

Index

157

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Tutorial, Release 10.4

Sage is free, open-source math software that supports research and teaching in algebra, geometry, number theory, cryptography, numerical computation, and related areas. Both the Sage development model and the technology in Sage itself are
distinguished by an extremely strong emphasis on openness, community, cooperation, and collaboration: we are building
the car, not reinventing the wheel. The overall goal of Sage is to create a viable, free, open-source alternative to Maple,
Mathematica, Magma, and MATLAB.
This tutorial is the best way to become familiar with Sage in only a few hours. You can read it in HTML or PDF versions,or from the Sage notebook (click Help, then click Tutorial to interactively work through the tutorial from within
Sage).
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.

CONTENTS

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2

CONTENTSCHAPTERONEINTRODUCTION

This tutorial should take at most 3-4 hours to fully work through. You can read it in HTML or PDF versions, or from theSage notebook click Help, then click Tutorial to interactively work through the tutorial from within Sage.
Though much of Sage is implemented using Python, no Python background is needed to read this tutorial. You will wantto learn Python (a very fun language!) at some point, and there are many excellent free resources for doing so: the Python
Beginner’s Guide [PyB] lists many options. If you just want to quickly try out Sage, this tutorial is the place to start. For
example:sage: 2 + 2
4
sage: factor(-2007)-1 * 3^2 * 223
sage: A = matrix(4,4, range(16)); A[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11][12 13 14 15]
sage: factor(A.charpoly())
x^2 * (x^2 - 30*x - 80)
sage: m = matrix(ZZ,2, range(4))
sage: m[0,0] = m[0,0] - 3sage: m[-3 1][ 2 3]
sage: E = EllipticCurve([1,2,3,4,5]);
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5
over Rational Field
sage: E.anlist(10)[0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3]
sage: E.rank()
1sage: k = 1/(sqrt(3)*I + 3/4 + sqrt(73)*5/9); k
36/(20*sqrt(73) + 36*I*sqrt(3) + 27)
sage: N(k)0.165495678130644 - 0.0521492082074256*I
sage: N(k,30)# 30 "bits"
0.16549568 - 0.052149208*I
sage: latex(k)(continues on next page)3

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(continued from previous page)\frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}>>> from sage.all import *>>> Integer(2) + Integer(2)4>>> factor(-Integer(2007))-1 * 3^2 * 223>>> A = matrix(Integer(4),Integer(4), range(Integer(16))); A[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11][12 13 14 15]>>> factor(A.charpoly())
x^2 * (x^2 - 30*x - 80)>>> m = matrix(ZZ,Integer(2), range(Integer(4)))>>> m[Integer(0),Integer(0)] = m[Integer(0),Integer(0)] - Integer(3)>>> m[-3 1][ 2 3]>>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]);>>> E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5
over Rational Field>>> E.anlist(Integer(10))[0, 1, 1, 0, -1, -3, 0, -1, -3, -3, -3]>>> E.rank()
1>>> k = Integer(1)/(sqrt(Integer(3))*I + Integer(3)/Integer(4) +␣˓→sqrt(Integer(73))*Integer(5)/Integer(9)); k
36/(20*sqrt(73) + 36*I*sqrt(3) + 27)>>> N(k)0.165495678130644 - 0.0521492082074256*I>>> N(k,Integer(30))# 30 "bits"
0.16549568 - 0.052149208*I>>> latex(k)\frac{36}{20 \, \sqrt{73} + 36 i \, \sqrt{3} + 27}

1.1 Installation
If you do not have Sage installed on a computer and just want to try some commands, use it online at http://sagecell.
sagemath.org.
See the Sage Installation Guide in the documentation section of the main Sage webpage [SA] for instructions on installing
Sage on your computer. Here we merely make a few comments.
1. The Sage download file comes with “batteries included”. In other words, although Sage uses Python, IPython,
PARI, GAP, Singular, Maxima, NTL, GMP, and so on, you do not need to install them separately as they areincluded with the Sage distribution. However, to use certain Sage features, e.g., Macaulay or KASH, you musthave the relevant programs installed on your computer already.

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2. The pre-compiled binary version of Sage (found on the Sage web site) may be easier and quicker to install than thesource code version. Just unpack the file and run sage.
3. If you’d like to use the SageTeX package (which allows you to embed the results of Sage computations into a LaTeX
file), you will need to make SageTeX known to your TeX distribution. To do this, see the section “Make SageTeX
known to TeX” in the Sage installation guide (this link should take you to a local copy of the installation guide).
It’s quite easy; you just need to set an environment variable or copy a single file to a directory that TeX will search.
The documentation for using SageTeX is located in $SAGE_ROOT/venv/share/texmf/tex/latex/
sagetex/, where “$SAGE_ROOT” refers to the directory where you installed Sage – for example, /opt/
sage-9.6.

1.2 Ways to Use Sage
You can use Sage in several ways.
• Notebook graphical interface: run sage -n jupyter; see the Jupyter documentation on-line,• Interactive command line: see The Interactive Shell,• Programs: By writing interpreted and compiled programs in Sage (see Loading and Attaching Sage files andCreating Compiled Code), and• Scripts: by writing stand-alone Python scripts that use the Sage library (see Standalone Python/Sage Scripts).

1.3 Longterm Goals for Sage
• Useful: Sage’s intended audience is mathematics students (from high school to graduate school), teachers, andresearch mathematicians. The aim is to provide software that can be used to explore and experiment with mathematical constructions in algebra, geometry, number theory, calculus, numerical computation, etc. Sage helps makeit easier to interactively experiment with mathematical objects.
• Efficient: Be fast. Sage uses highly-optimized mature software like GMP, PARI, GAP, and NTL, and so is very
fast at certain operations.
• Free and open source: The source code must be freely available and readable, so users can understand what thesystem is really doing and more easily extend it. Just as mathematicians gain a deeper understanding of a theorem
by carefully reading or at least skimming the proof, people who do computations should be able to understand how
the calculations work by reading documented source code. If you use Sage to do computations in a paper you
publish, you can rest assured that your readers will always have free access to Sage and all its source code, and you
are even allowed to archive and re-distribute the version of Sage you used.
• Easy to compile: Sage should be easy to compile from source for Linux, OS X and Windows users. This provides
more flexibility for users to modify the system.
• Cooperation: Provide robust interfaces to most other computer algebra systems, including PARI, GAP, Singular,
Maxima, KASH, Magma, Maple, and Mathematica. Sage is meant to unify and extend existing math software.
• Well documented: Tutorial, programming guide, reference manual, and how-to, with numerous examples anddiscussion of background mathematics.
• Extensible: Be able to define new data types or derive from built-in types, and use code written in a range oflanguages.
• User friendly: It should be easy to understand what functionality is provided for a given object and to view documentation and source code. Also attain a high level of user support.

1.2. Ways to Use Sage

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Chapter 1. Introduction

CHAPTERTWO

A GUIDED TOUR

This section is a guided tour of some of what is available in Sage. For many more examples, see “Sage Constructions”,
which is intended to answer the general question “How do I construct …?”. See also the “Sage Reference Manual”, which
has thousands more examples. Also note that you can interactively work through this tour in the Sage notebook by clicking
the Help link.
(If you are viewing the tutorial in the Sage notebook, press shift-enter to evaluate any input cell. You can even edit
the input before pressing shift-enter. On some Macs you might have to press shift-return rather than shift-enter.)

2.1 Assignment, Equality, and Arithmetic
With some minor exceptions, Sage uses the Python programming language, so most introductory books on Python will
help you to learn Sage.
Sage uses = for assignment. It uses ==, <=, >=, < and > for comparison:
sage: a = 5
sage: a5
sage: 2 == 2
True
sage: 2 == 3
False
sage: 2 < 3Truesage: a == 5
True>>> from sage.all import *>>> a = Integer(5)>>> a5>>> Integer(2) == Integer(2)True>>> Integer(2) == Integer(3)False>>> Integer(2) < Integer(3)True>>> a == Integer(5)TrueSage provides all of the basic mathematical operations:7

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sage: 2**3
# ** means exponent
8
sage: 2^3
# ^ is a synonym for ** (unlike in Python)
8
sage: 10 % 3 # for integer arguments, % means mod, i.e., remainder
1
sage: 10/4
5/2
sage: 10//4# for integer arguments, // returns the integer quotient2
sage: 4 * (10 // 4) + 10 % 4 == 10
True
sage: 3^2*4 + 2%538>>> from sage.all import *>>> Integer(2)**Integer(3)# ** means exponent
8>>> Integer(2)**Integer(3)# ^ is a synonym for ** (unlike in Python)
8>>> Integer(10) % Integer(3) # for integer arguments, % means mod, i.e., remainder
1>>> Integer(10)/Integer(4)5/2>>> Integer(10)//Integer(4)# for integer arguments, // returns the integer␣˓→quotient2>>> Integer(4) * (Integer(10) // Integer(4)) + Integer(10) % Integer(4) == Integer(10)True>>> Integer(3)**Integer(2)*Integer(4) + Integer(2)%Integer(5)38The computation of an expression like 3^2*4 + 2%5 depends on the order in which the operations are applied; this isspecified in the “operator precedence table” in Arithmetical binary operator precedence.
Sage also provides many familiar mathematical functions; here are just a few examples:sage: sqrt(3.4)
1.84390889145858
sage: sin(5.135)-0.912021158525540
sage: sin(pi/3)
1/2*sqrt(3)>>> from sage.all import *>>> sqrt(RealNumber( 3.4 ))1.84390889145858>>> sin(RealNumber( 5.135 ))-0.912021158525540>>> sin(pi/Integer(3))
1/2*sqrt(3)As the last example shows, some mathematical expressions return ‘exact’ values, rather than numerical approximations.
To get a numerical approximation, use either the function N or the method n (and both of these have a longer name,numerical_approx, and the function N is the same as n)). These take optional arguments prec, which is therequested number of bits of precision, and digits, which is the requested number of decimal digits of precision; thedefault is 53 bits of precision.
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sage: exp(2)e^2sage: n(exp(2))
7.38905609893065
sage: sqrt(pi).numerical_approx()1.77245385090552
sage: sin(10).n(digits=5)-0.54402
sage: N(sin(10),digits=10)-0.5440211109
sage: numerical_approx(pi, prec=200)
3.1415926535897932384626433832795028841971693993751058209749>>> from sage.all import *>>> exp(Integer(2))
e^2>>> n(exp(Integer(2)))
7.38905609893065>>> sqrt(pi).numerical_approx()1.77245385090552>>> sin(Integer(10)).n(digits=Integer(5))-0.54402>>> N(sin(Integer(10)),digits=Integer(10))-0.5440211109>>> numerical_approx(pi, prec=Integer(200))
3.1415926535897932384626433832795028841971693993751058209749

Python is dynamically typed, so the value referred to by each variable has a type associated with it, but a given variable
may hold values of any Python type within a given scope:
sage: a = 5# a is an integer
sage: type(a)<class sage.rings.integer.Integer >sage: a = 5/3 # now a is a rational number
sage: type(a)<class sage.rings.rational.Rational >sage: a = hello# now a is a string
sage: type(a)<... str >>>> from sage.all import *>>> a = Integer(5)# a is an integer>>> type(a)<class sage.rings.integer.Integer >>>> a = Integer(5)/Integer(3) # now a is a rational number>>> type(a)<class sage.rings.rational.Rational >>>> a = hello# now a is a string>>> type(a)<... str >The C programming language, which is statically typed, is much different; a variable declared to hold an int can only holdan int in its scope.

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2.2 Getting Help
Sage has extensive built-in documentation, accessible by typing the name of a function or a constant (for example),followed by a question mark:sage: tan?
Type:
Definition:
Docstring:<class sage.calculus.calculus.Function_tan >tan( [noargspec] )The tangent function
EXAMPLES:
sage: tan(pi)0sage: tan(3.1415)-0.0000926535900581913
sage: tan(3.1415/4)
0.999953674278156
sage: tan(pi/4)
1
sage: tan(1/2)
tan(1/2)
sage: RR(tan(1/2))
0.546302489843790
sage: log2?
Type:<class sage.functions.constants.Log2 >Definition: log2( [noargspec] )Docstring:
The natural logarithm of the real number 2.
EXAMPLES:
sage: log2
log2
sage: float(log2)
0.69314718055994529
sage: RR(log2)
0.693147180559945
sage: R = RealField(200); R
Real Field with 200 bits of precision
sage: R(log2)
0.69314718055994530941723212145817656807550013436025525412068
sage: l = (1-log2)/(1+log2); l(1 - log(2))/(log(2) + 1)
sage: R(l)0.18123221829928249948761381864650311423330609774776013488056
sage: maxima(log2)log(2)
sage: maxima(log2).float().6931471805599453
sage: gp(log2)
0.6931471805599453094172321215# 32-bit
0.69314718055994530941723212145817656807# 64-bitsage: sudoku?
File:sage/local/lib/python2.5/site-packages/sage/games/sudoku.pyType:<... function >(continues on next page)10

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(continued from previous page)Definition:
Docstring:sudoku(A)Solve the 9x9 Sudoku puzzle defined by the matrix A.
EXAMPLE:sage: A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0,
0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0,
0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0,
0,3,0, 0,0,2,
0,0,0, 4,9,0, 0,5,0, 0,0,3])
sage: A
[5 0 0 0 8 0 0 4 9]
[0 0 0 5 0 0 0 3 0]
[0 6 7 3 0 0 0 0 1]
[1 5 0 0 0 0 0 0 0]
[0 0 0 2 0 8 0 0 0]
[0 0 0 0 0 0 0 1 8]
[7 0 0 0 0 4 1 5 0]
[0 3 0 0 0 2 0 0 0]
[4 9 0 0 5 0 0 0 3]
sage: sudoku(A)[5 1 3 6 8 7 2 4 9]
[8 4 9 5 2 1 6 3 7]
[2 6 7 3 4 9 5 8 1]
[1 5 8 4 6 3 9 7 2]
[9 7 4 2 1 8 3 6 5]
[3 2 6 7 9 5 4 1 8]
[7 8 2 9 3 4 1 5 6]
[6 3 5 1 7 2 8 9 4]
[4 9 1 8 5 6 7 2 3]>>> from sage.all import *>>> tan?
Type:<class sage.calculus.calculus.Function_tan >Definition: tan( [noargspec] )Docstring:
The tangent function
EXAMPLES:>>> tan(pi)0>>> tan(RealNumber( 3.1415 ))-0.0000926535900581913>>> tan(RealNumber( 3.1415 )/Integer(4))
0.999953674278156>>> tan(pi/Integer(4))
1>>> tan(Integer(1)/Integer(2))
tan(1/2)>>> RR(tan(Integer(1)/Integer(2)))
0.546302489843790>>> log2?
Type:<class sage.functions.constants.Log2 >Definition: log2( [noargspec] )Docstring:(continues on next page)2.2. Getting Help11

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(continued from previous page)The natural logarithm of the real number 2.
EXAMPLES:>>> log2
log2>>> float(log2)
0.69314718055994529>>> RR(log2)
0.693147180559945>>> R = RealField(Integer(200)); R
Real Field with 200 bits of precision>>> R(log2)
0.69314718055994530941723212145817656807550013436025525412068>>> l = (Integer(1)-log2)/(Integer(1)+log2); l(1 - log(2))/(log(2) + 1)>>> R(l)0.18123221829928249948761381864650311423330609774776013488056>>> maxima(log2)log(2)>>> maxima(log2).float().6931471805599453>>> gp(log2)
0.6931471805599453094172321215# 32-bit
0.69314718055994530941723212145817656807# 64-bit>>> sudoku?
File:sage/local/lib/python2.5/site-packages/sage/games/sudoku.pyType:<... function >Definition: sudoku(A)Docstring:
Solve the 9x9 Sudoku puzzle defined by the matrix A.
EXAMPLE:>>> A = matrix(ZZ,Integer(9),[Integer(5),Integer(0),Integer(0), Integer(0),Integer(8),˓→Integer(0), Integer(0),Integer(4),Integer(9), Integer(0),Integer(0),Integer(0),␣
˓→Integer(5),Integer(0),Integer(0),0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0,
0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0,
0,3,0, 0,0,2,
0,0,0, 4,9,0, 0,5,0, 0,0,3])>>> A[5 0 0 0 8 0 0 4 9]
[0 0 0 5 0 0 0 3 0]
[0 6 7 3 0 0 0 0 1]
[1 5 0 0 0 0 0 0 0]
[0 0 0 2 0 8 0 0 0]
[0 0 0 0 0 0 0 1 8]
[7 0 0 0 0 4 1 5 0]
[0 3 0 0 0 2 0 0 0]
[4 9 0 0 5 0 0 0 3]>>> sudoku(A)[5 1 3 6 8 7 2 4 9]
[8 4 9 5 2 1 6 3 7]
[2 6 7 3 4 9 5 8 1]
[1 5 8 4 6 3 9 7 2]
[9 7 4 2 1 8 3 6 5]
[3 2 6 7 9 5 4 1 8]
(continues on next page)

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(continued from previous page)[7 8 2 9 3 4 1 5 6]
[6 3 5 1 7 2 8 9 4]
[4 9 1 8 5 6 7 2 3]

Sage also provides ‘Tab completion’: type the first few letters of a function and then hit the Tab key. For example, if you
type ta followed by Tab, Sage will print tachyon, tan, tanh, taylor. This provides a good way to find thenames of functions and other structures in Sage.

2.3 Functions, Indentation, and Counting
To define a new function in Sage, use the def command and a colon after the list of variable names. For example:
sage: def is_even(n):....:return n%2 == 0
sage: is_even(2)True
sage: is_even(3)False>>> from sage.all import *>>> def is_even(n):...
return n%Integer(2) == Integer(0)>>> is_even(Integer(2))
True>>> is_even(Integer(3))
False

Note: Depending on which version of the tutorial you are viewing, you may see three dots ....: on the second line ofthis example. Do not type them; they are just to emphasize that the code is indented. Whenever this is the case, press[Return/Enter] once at the end of the block to insert a blank line and conclude the function definition.
You do not specify the types of any of the input arguments. You can specify multiple inputs, each of which may have an
optional default value. For example, the function below defaults to divisor=2 if divisor is not specified.
sage: def is_divisible_by(number, divisor=2):....:
return number%divisor == 0
sage: is_divisible_by(6,2)
True
sage: is_divisible_by(6)Truesage: is_divisible_by(6, 5)
False>>> from sage.all import *>>> def is_divisible_by(number, divisor=Integer(2)):...
return number%divisor == Integer(0)>>> is_divisible_by(Integer(6),Integer(2))
True>>> is_divisible_by(Integer(6))
True>>> is_divisible_by(Integer(6), Integer(5))
False

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You can also explicitly specify one or either of the inputs when calling the function; if you specify the inputs explicitly,
you can give them in any order:
sage: is_divisible_by(6, divisor=5)
False
sage: is_divisible_by(divisor=2, number=6)
True>>> from sage.all import *>>> is_divisible_by(Integer(6), divisor=Integer(5))
False>>> is_divisible_by(divisor=Integer(2), number=Integer(6))
True

In Python, blocks of code are not indicated by curly braces or begin and end blocks as in many other languages. Instead,blocks of code are indicated by indentation, which must match up exactly. For example, the following is a syntax error
because the return statement is not indented the same amount as the other lines above it.
sage: def even(n):....:v = []....:
for i in range(3,n):....:if i % 2 == 0:....:
v.append(i)....:return v
Syntax Error:
return v>>> from sage.all import *>>> def even(n):...
v = []...
for i in range(Integer(3),n):...
if i % Integer(2) == Integer(0):...
v.append(i)...
return v
Syntax Error:
return v

If you fix the indentation, the function works:
sage: def even(n):....:v = []....:
for i in range(3,n):....:if i % 2 == 0:....:
v.append(i)....:return v
sage: even(10)[4, 6, 8]>>> from sage.all import *>>> def even(n):...
v = []...
for i in range(Integer(3),n):...
if i % Integer(2) == Integer(0):...
v.append(i)...
return v>>> even(Integer(10))[4, 6, 8]

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Semicolons are not needed at the ends of lines; a line is in most cases ended by a newline. However, you can put multiple
statements on one line, separated by semicolons:
sage: a = 5; b = a + 3; c = b^2; c
64>>> from sage.all import *>>> a = Integer(5); b = a + Integer(3); c = b**Integer(2); c
64

If you would like a single line of code to span multiple lines, use a terminating backslash:sage: 2 + \
....:
3
5
>>> from sage.all import *>>> Integer(2) +Integer(3)5In Sage, you count by iterating over a range of integers. For example, the first line below is exactly like for(i=0;i<3; i++) in C++ or Java:sage: for i in range(3):....:print(i)01
2>>> from sage.all import *>>> for i in range(Integer(3)):...
print(i)01
2

The first line below is like for(i=2;i<5;i++).
sage: for i in range(2,5):....:
print(i)234>>> from sage.all import *>>> for i in range(Integer(2),Integer(5)):...
print(i)2
34

The third argument controls the step, so the following is like for(i=1;i<6;i+=2).
sage: for i in range(1,6,2):....:
print(i)1(continues on next page)2.3. Functions, Indentation, and Counting

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(continued from previous page)3
5>>> from sage.all import *>>> for i in range(Integer(1),Integer(6),Integer(2)):...
print(i)135Often you will want to create a nice table to display numbers you have computed using Sage. One easy way to do this is
to use string formatting. Below, we create three columns each of width exactly 6 and make a table of squares and cubes.
sage: for i in range(5):....:print( %6s %6s %6s
0
0
0
1
1
1
2
4
8
3
9
27
4
16
64

% (i, i^2, i^3))>>> from sage.all import *>>> for i in range(Integer(5)):...
print( %6s %6s %6s % (i, i**Integer(2), i**Integer(3)))
0
0
0
1
1
1
2
4
8
3
9
27
4
16
64

The most basic data structure in Sage is the list, which is – as the name suggests – just a list of arbitrary objects. For
example, using range, the following command creates a list:
sage: list(range(2,10))[2, 3, 4, 5, 6, 7, 8, 9]>>> from sage.all import *>>> list(range(Integer(2),Integer(10)))[2, 3, 4, 5, 6, 7, 8, 9]Here is a more complicated list:
sage: v = [1, "hello", 2/3, sin(x^3)]
sage: v
[1, hello , 2/3, sin(x^3)]>>> from sage.all import *>>> v = [Integer(1), "hello", Integer(2)/Integer(3), sin(x**Integer(3))]>>> v[1, hello , 2/3, sin(x^3)]

List indexing is 0-based, as in many programming languages.
sage: v[0]1(continues on next page)

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(continued from previous page)

sage: v[3]sin(x^3)>>> from sage.all import *>>> v[Integer(0)]1>>> v[Integer(3)]sin(x^3)

Use len(v) to get the length of v, use v.append(obj) to append a new object to the end of v, and use del v[i]to delete the 𝑖𝑡ℎ entry of v:
sage: len(v)4sage: v.append(1.5)
sage: v
[1, hello , 2/3, sin(x^3), 1.50000000000000]
sage: del v[1]sage: v
[1, 2/3, sin(x^3), 1.50000000000000]>>> from sage.all import *>>> len(v)4>>> v.append(RealNumber( 1.5 ))>>> v[1, hello , 2/3, sin(x^3), 1.50000000000000]>>> del v[Integer(1)]>>> v[1, 2/3, sin(x^3), 1.50000000000000]

Another important data structure is the dictionary (or associative array). This works like a list, except that it can beindexed with almost any object (the indices must be immutable):sage: d = { hi :-2,
sage: d[ hi ]-2sage: d[e]pi

3/8:pi,>>> from sage.all import *>>> d = { hi :-Integer(2),>>> d[ hi ]-2>>> d[e]pie:pi}Integer(3)/Integer(8):pi,e:pi}

You can also define new data types using classes. Encapsulating mathematical objects with classes is a powerful techniquethat can help to simplify and organize your Sage programs. Below, we define a class that represents the list of even positive
integers up to n; it derives from the builtin type list.
sage: class Evens(list):....:
def __init__(self, n):....:self.n = n....:list.__init__(self, range(2, n+1, 2))(continues on next page)2.3. Functions, Indentation, and Counting

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(continued from previous page)

....:
....:

def __repr__(self):
return "Even positive numbers up to n.">>> from sage.all import *>>> class Evens(list):...
def __init__(self, n):...
self.n = n...
list.__init__(self, range(Integer(2), n+Integer(1), Integer(2)))...
def __repr__(self):...
return "Even positive numbers up to n."

The __init__ method is called to initialize the object when it is created; the __repr__ method prints the object out.
We call the list constructor method in the second line of the __init__ method. We create an object of class Evens
as follows:
sage: e = Evens(10)sage: eEven positive numbers up to n.
>>> from sage.all import *>>> e = Evens(Integer(10))>>> eEven positive numbers up to n.

Note that e prints using the __repr__ method that we defined. To see the underlying list of numbers, use the list
function:sage: list(e)[2, 4, 6, 8, 10]>>> from sage.all import *>>> list(e)[2, 4, 6, 8, 10]

We can also access the n attribute or treat e like a list.
sage: e.n10sage: e[2]6>>> from sage.all import *>>> e.n10>>> e[Integer(2)]618

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2.4 Basic Algebra and Calculus
Sage can perform various computations related to basic algebra and calculus: for example, finding solutions to equations,differentiation, integration, and Laplace transforms. See the Sage Constructions documentation for more examples.
In all these examples, it is important to note that the variables in the functions are defined to be var(...). As an
example:sage: u = var( u )sage: diff(sin(u), u)cos(u)>>> from sage.all import *>>> u = var( u )>>> diff(sin(u), u)cos(u)

If you get a NameError, check to see if you misspelled something, or forgot to define a variable with var(...).

2.4.1 Solving Equations
Solving Equations Exactly
The solve function solves equations. To use it, first specify some variables; then the arguments to solve are an
equation (or a system of equations), together with the variables for which to solve:
sage: x = var( x )sage: solve(x^2 + 3*x + 2, x)[x == -2, x == -1]>>> from sage.all import *>>> x = var( x )>>> solve(x**Integer(2) + Integer(3)*x + Integer(2), x)[x == -2, x == -1]

You can solve equations for one variable in terms of others:
sage: x, b, c = var( x b c )sage: solve([x^2 + b*x + c == 0],x)[x == -1/2*b - 1/2*sqrt(b^2 - 4*c), x == -1/2*b + 1/2*sqrt(b^2 - 4*c)]>>> from sage.all import *>>> x, b, c = var( x b c )>>> solve([x**Integer(2) + b*x + c == Integer(0)],x)[x == -1/2*b - 1/2*sqrt(b^2 - 4*c), x == -1/2*b + 1/2*sqrt(b^2 - 4*c)]

You can also solve for several variables:sage: x, y = var( x, y )sage: solve([x+y==6, x-y==4], x, y)[[x == 5, y == 1]]

2.4. Basic Algebra and Calculus

19Tutorial, Release 10.4>>> from sage.all import *>>> x, y = var( x, y )>>> solve([x+y==Integer(6), x-y==Integer(4)], x, y)[[x == 5, y == 1]]

The following example of using Sage to solve a system of non-linear equations was provided by Jason Grout: first, we
solve the system symbolically:
sage: var( x y p q )(x, y, p, q)
sage: eq1 = p+q==9
sage: eq2 = q*y+p*x==-6
sage: eq3 = q*y^2+p*x^2==24
sage: solve([eq1,eq2,eq3,p==1],p,q,x,y)[[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(10) - 2/3], [p == 1, q == 8,
˓→ x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(10) - 2/3]]>>> from sage.all import *>>> var( x y p q )(x, y, p, q)>>> eq1 = p+q==Integer(9)>>> eq2 = q*y+p*x==-Integer(6)>>> eq3 = q*y**Integer(2)+p*x**Integer(2)==Integer(24)>>> solve([eq1,eq2,eq3,p==Integer(1)],p,q,x,y)[[p == 1, q == 8, x == -4/3*sqrt(10) - 2/3, y == 1/6*sqrt(10) - 2/3], [p == 1, q == 8,
˓→ x == 4/3*sqrt(10) - 2/3, y == -1/6*sqrt(10) - 2/3]]

For numerical approximations of the solutions, you can instead use:
sage: solns = solve([eq1,eq2,eq3,p==1],p,q,x,y, solution_dict=True)
sage: [[s[p].n(30), s[q].n(30), s[x].n(30), s[y].n(30)] for s in solns][[1.0000000, 8.0000000, -4.8830369, -0.13962039],[1.0000000, 8.0000000, 3.5497035, -1.1937129]]>>> from sage.all import *>>> solns = solve([eq1,eq2,eq3,p==Integer(1)],p,q,x,y, solution_dict=True)>>> [[s[p].n(Integer(30)), s[q].n(Integer(30)), s[x].n(Integer(30)), s[y].
˓→n(Integer(30))] for s in solns][[1.0000000, 8.0000000, -4.8830369, -0.13962039],[1.0000000, 8.0000000, 3.5497035, -1.1937129]](The function n prints a numerical approximation, and the argument is the number of bits of precision.)
Solving Equations NumericallyOften times, solve will not be able to find an exact solution to the equation or equations specified. When it fails, you can
use find_root to find a numerical solution. For example, solve does not return anything interesting for the following
equation:sage: theta = var( theta )sage: solve(cos(theta)==sin(theta), theta)[sin(theta) == cos(theta)]>>> from sage.all import *>>> theta = var( theta )(continues on next page)

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(continued from previous page)>>> solve(cos(theta)==sin(theta), theta)[sin(theta) == cos(theta)]

On the other hand, we can use find_root to find a solution to the above equation in the range 0 < 𝜑 < 𝜋/2:
sage: phi = var( phi )sage: find_root(cos(phi)==sin(phi),0,pi/2)
0.785398163397448...
>>> from sage.all import *>>> phi = var( phi )>>> find_root(cos(phi)==sin(phi),Integer(0),pi/Integer(2))
0.785398163397448...

2.4.2 Differentiation, Integration, etc.
Sage knows how to differentiate and integrate many functions. For example, to differentiate sin(𝑢) with respect to 𝑢, dothe following:
sage: u = var( u )sage: diff(sin(u), u)cos(u)>>> from sage.all import *>>> u = var( u )>>> diff(sin(u), u)cos(u)

To compute the fourth derivative of sin(𝑥2 ):sage: diff(sin(x^2), x, 4)
16*x^4*sin(x^2) - 48*x^2*cos(x^2) - 12*sin(x^2)>>> from sage.all import *>>> diff(sin(x**Integer(2)), x, Integer(4))
16*x^4*sin(x^2) - 48*x^2*cos(x^2) - 12*sin(x^2)

To compute the partial derivatives of 𝑥2 + 17𝑦 2 with respect to 𝑥 and 𝑦, respectively:
sage: x, y = var( x,y )sage: f = x^2 + 17*y^2
sage: f.diff(x)2*xsage: f.diff(y)34*y>>> from sage.all import *>>> x, y = var( x,y )>>> f = x**Integer(2) + Integer(17)*y**Integer(2)>>> f.diff(x)2*x>>> f.diff(y)34*y2.4. Basic Algebra and Calculus

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We move on to integrals, both indefinite and definite. To compute∫︀

𝑥 sin(𝑥2 ) 𝑑𝑥 and

∫︀ 1

𝑥
𝑑𝑥
0 𝑥2 +1

sage: integral(x*sin(x^2), x)-1/2*cos(x^2)
sage: integral(x/(x^2+1), x, 0, 1)
1/2*log(2)>>> from sage.all import *>>> integral(x*sin(x**Integer(2)), x)-1/2*cos(x^2)>>> integral(x/(x**Integer(2)+Integer(1)), x, Integer(0), Integer(1))
1/2*log(2)To compute the partial fraction decomposition of 𝑥21−1 :sage: f = 1/((1+x)*(x-1))
sage: f.partial_fraction(x)-1/2/(x + 1) + 1/2/(x - 1)>>> from sage.all import *>>> f = Integer(1)/((Integer(1)+x)*(x-Integer(1)))>>> f.partial_fraction(x)-1/2/(x + 1) + 1/2/(x - 1)

2.4.3 Solving Differential Equations
You can use Sage to investigate ordinary differential equations. To solve the equation 𝑥′ + 𝑥 − 1 = 0:
sage: t = var( t )# define a variable t
sage: x = function( x )(t)# define x to be a function of that variable
sage: DE = diff(x, t) + x - 1
sage: desolve(DE, [x,t])(_C + e^t)*e^(-t)>>> from sage.all import *>>> t = var( t )# define a variable t>>> x = function( x )(t)# define x to be a function of that variable>>> DE = diff(x, t) + x - Integer(1)>>> desolve(DE, [x,t])(_C + e^t)*e^(-t)

This uses Sage’s interface to Maxima [Max], and so its output may be a bit different from other Sage output. In this case,this says that the general solution to the differential equation is 𝑥(𝑡) = 𝑒−𝑡 (𝑒𝑡 + 𝑐).
You can compute Laplace transforms also; the Laplace transform of 𝑡2 𝑒𝑡 − sin(𝑡) is computed as follows:
sage: s = var("s")
sage: t = var("t")
sage: f = t^2*exp(t) - sin(t)sage: f.laplace(t,s)-1/(s^2 + 1) + 2/(s - 1)^3>>> from sage.all import *>>> s = var("s")>>> t = var("t")(continues on next page)

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(continued from previous page)>>> f = t**Integer(2)*exp(t) - sin(t)>>> f.laplace(t,s)-1/(s^2 + 1) + 2/(s - 1)^3

Here is a more involved example. The displacement from equilibrium (respectively) for a coupled spring attached to awall on the left|------\/\/\/\/\---|mass1|----\/\/\/\/\/----|mass2|
spring1
spring2

is modeled by the system of 2nd order differential equations
𝑚1 𝑥′′1 + (𝑘1 + 𝑘2 )𝑥1 − 𝑘2 𝑥2 = 0𝑚2 𝑥′′2 + 𝑘2 (𝑥2 − 𝑥1 ) = 0,
where 𝑚𝑖 is the mass of object i, 𝑥𝑖 is the displacement from equilibrium of mass i, and 𝑘𝑖 is the spring constant for
spring i.
Example: Use Sage to solve the above problem with 𝑚1 = 2, 𝑚2 = 1, 𝑘1 = 4, 𝑘2 = 2, 𝑥1 (0) = 3, 𝑥′1 (0) = 0,𝑥2 (0) = 3, 𝑥′2 (0) = 0.
Solution: Take the Laplace transform of the first equation (with the notation 𝑥 = 𝑥1 , 𝑦 = 𝑥2 ):sage: t,s = SR.var( t,s )sage: x = function( x )sage: y = function( y )sage: f = 2*x(t).diff(t,2) + 6*x(t) - 2*y(t)sage: f.laplace(t,s)
2*s^2*laplace(x(t), t, s) - 2*s*x(0) + 6*laplace(x(t), t, s) - 2*laplace(y(t), t, s) ˓→ 2*D[0](x)(0)>>> from sage.all import *>>> t,s = SR.var( t,s )>>> x = function( x )>>> y = function( y )>>> f = Integer(2)*x(t).diff(t,Integer(2)) + Integer(6)*x(t) - Integer(2)*y(t)>>> f.laplace(t,s)
2*s^2*laplace(x(t), t, s) - 2*s*x(0) + 6*laplace(x(t), t, s) - 2*laplace(y(t), t, s) ˓→ 2*D[0](x)(0)This is hard to read, but it says that
−2𝑥′ (0) + 2𝑠2 · 𝑋(𝑠) − 2𝑠𝑥(0) − 2𝑌 (𝑠) + 6𝑋(𝑠) = 0(where the Laplace transform of a lower case function like 𝑥(𝑡) is the upper case function 𝑋(𝑠)). Take the Laplacetransform of the second equation:
sage: de2 = maxima("diff(y(t),t, 2) + 2*y(t) - 2*x(t)")
sage: lde2 = de2.laplace("t","s"); lde2.sage()s^2*laplace(y(t), t, s) - s*y(0) - 2*laplace(x(t), t, s) + 2*laplace(y(t), t, s) -␣˓→D[0](y)(0)>>> from sage.all import *>>> de2 = maxima("diff(y(t),t, 2) + 2*y(t) - 2*x(t)")>>> lde2 = de2.laplace("t","s"); lde2.sage()s^2*laplace(y(t), t, s) - s*y(0) - 2*laplace(x(t), t, s) + 2*laplace(y(t), t, s) -␣˓→D[0](y)(0)2.4. Basic Algebra and Calculus

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This says−𝑌 ′ (0) + 𝑠2 𝑌 (𝑠) + 2𝑌 (𝑠) − 2𝑋(𝑠) − 𝑠𝑦(0) = 0.
Plug in the initial conditions for 𝑥(0), 𝑥′ (0), 𝑦(0), and 𝑦 ′ (0), and solve the resulting two equations:sage: var( s X Y )(s, X, Y)
sage: eqns = [(2*s^2+6)*X-2*Y == 6*s, -2*X +(s^2+2)*Y == 3*s]
sage: solve(eqns, X,Y)[[X == 3*(s^3 + 3*s)/(s^4 + 5*s^2 + 4),
Y == 3*(s^3 + 5*s)/(s^4 + 5*s^2 + 4)]]>>> from sage.all import *>>> var( s X Y )(s, X, Y)>>> eqns = [(Integer(2)*s**Integer(2)+Integer(6))*X-Integer(2)*Y == Integer(6)*s, ˓→Integer(2)*X +(s**Integer(2)+Integer(2))*Y == Integer(3)*s]>>> solve(eqns, X,Y)[[X == 3*(s^3 + 3*s)/(s^4 + 5*s^2 + 4),
Y == 3*(s^3 + 5*s)/(s^4 + 5*s^2 + 4)]]

Now take inverse Laplace transforms to get the answer:
sage: var( s t )(s, t)
sage: inverse_laplace((3*s^3 + 9*s)/(s^4 + 5*s^2 + 4),s,t)cos(2*t) + 2*cos(t)sage: inverse_laplace((3*s^3 + 15*s)/(s^4 + 5*s^2 + 4),s,t)-cos(2*t) + 4*cos(t)>>> from sage.all import *>>> var( s t )(s, t)>>> inverse_laplace((Integer(3)*s**Integer(3) + Integer(9)*s)/(s**Integer(4) +␣˓→Integer(5)*s**Integer(2) + Integer(4)),s,t)cos(2*t) + 2*cos(t)>>> inverse_laplace((Integer(3)*s**Integer(3) + Integer(15)*s)/(s**Integer(4) +␣˓→Integer(5)*s**Integer(2) + Integer(4)),s,t)-cos(2*t) + 4*cos(t)Therefore, the solution is𝑥1 (𝑡) = cos(2𝑡) + 2 cos(𝑡),𝑥2 (𝑡) = 4 cos(𝑡) − cos(2𝑡).

This can be plotted parametrically using
sage: t = var( t )sage: P = parametric_plot((cos(2*t) + 2*cos(t), 4*cos(t) - cos(2*t) ),....:(t, 0, 2*pi), rgbcolor=hue(0.9))
sage: show(P)>>> from sage.all import *>>> t = var( t )>>> P = parametric_plot((cos(Integer(2)*t) + Integer(2)*cos(t), Integer(4)*cos(t) -␣˓→cos(Integer(2)*t) ),...
(t, Integer(0), Integer(2)*pi), rgbcolor=hue(RealNumber( 0.9 )))>>> show(P)24

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The individual components can be plotted using
sage: t = var( t )sage: p1 = plot(cos(2*t) + 2*cos(t), (t,0, 2*pi), rgbcolor=hue(0.3))
sage: p2 = plot(4*cos(t) - cos(2*t), (t,0, 2*pi), rgbcolor=hue(0.6))
sage: show(p1 + p2)>>> from sage.all import *>>> t = var( t )>>> p1 = plot(cos(Integer(2)*t) + Integer(2)*cos(t), (t,Integer(0), Integer(2)*pi),␣
˓→rgbcolor=hue(RealNumber( 0.3 )))>>> p2 = plot(Integer(4)*cos(t) - cos(Integer(2)*t), (t,Integer(0), Integer(2)*pi),␣
˓→rgbcolor=hue(RealNumber( 0.6 )))>>> show(p1 + p2)

For more on plotting, see Plotting. See section 5.5 of [NagleEtAl2004] for further information on differential equations.

2.4.4 Euler’s Method for Systems of Differential Equations
In the next example, we will illustrate Euler’s method for first and second order ODEs. We first recall the basic idea forfirst order equations. Given an initial value problem of the form𝑦 ′ = 𝑓 (𝑥, 𝑦),𝑦(𝑎) = 𝑐,we want to find the approximate value of the solution at 𝑥 = 𝑏 with 𝑏 > 𝑎.
Recall from the definition of the derivative that𝑦 ′ (𝑥) ≈𝑦(𝑥 + ℎ) − 𝑦(𝑥),ℎwhere ℎ > 0 is given and small. This and the DE together give 𝑓 (𝑥, 𝑦(𝑥)) ≈ 𝑦(𝑥+ℎ)−𝑦(𝑥). Now solve for 𝑦(𝑥 + ℎ):ℎ𝑦(𝑥 + ℎ) ≈ 𝑦(𝑥) + ℎ · 𝑓 (𝑥, 𝑦(𝑥)).
If we call ℎ·𝑓 (𝑥, 𝑦(𝑥)) the “correction term” (for lack of anything better), call 𝑦(𝑥) the “old value of 𝑦”, and call 𝑦(𝑥+ℎ)the “new value of 𝑦”, then this approximation can be re-expressed as
𝑦𝑛𝑒𝑤 ≈ 𝑦𝑜𝑙𝑑 + ℎ · 𝑓 (𝑥, 𝑦𝑜𝑙𝑑 ).
If we break the interval from 𝑎 to 𝑏 into 𝑛 steps, so that ℎ = 𝑏−𝑎𝑛 , then we can record the information for this method ina table.
𝑥𝑦ℎ · 𝑓 (𝑥, 𝑦)𝑎𝑎+ℎ𝑎 + 2ℎ…
𝑏 = 𝑎 + 𝑛ℎ𝑐𝑐 + ℎ · 𝑓 (𝑎, 𝑐)…

ℎ · 𝑓 (𝑎, 𝑐)
…

???

…

The goal is to fill out all the blanks of the table, one row at a time, until we reach the ??? entry, which is the Euler’s
method approximation for 𝑦(𝑏).
The idea for systems of ODEs is similar.

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Example: Numerically approximate 𝑧(𝑡) at 𝑡 = 1 using 4 steps of Euler’s method, where 𝑧 ′′ + 𝑡𝑧 ′ + 𝑧 = 0, 𝑧(0) = 1,𝑧 ′ (0) = 0.
We must reduce the 2nd order ODE down to a system of two first order DEs (using 𝑥 = 𝑧, 𝑦 = 𝑧 ′ ) and apply Euler’s
method:sage: t,x,y = PolynomialRing(RealField(10),3,"txy").gens()
sage: f = y; g = -x - y * t
sage: eulers_method_2x2(f,g, 0, 1, 0, 1/4, 1)tx
h*f(t,x,y)
y
0
1
0.00
0
1/4
1.0
-0.062
-0.25
1/2
0.94
-0.12
-0.48
3/4
0.82
-0.16
-0.66
1
0.65
-0.18
-0.74

h*g(t,x,y)
-0.25
-0.23
-0.17
-0.081
0.022>>> from sage.all import *>>> t,x,y = PolynomialRing(RealField(Integer(10)),Integer(3),"txy").gens()>>> f = y; g = -x - y * t>>> eulers_method_2x2(f,g, Integer(0), Integer(1), Integer(0), Integer(1)/Integer(4),␣
˓→Integer(1))
txh*f(t,x,y)yh*g(t,x,y)
0
1
0.00
0
-0.25
1/4
1.0
-0.062
-0.25
-0.23
1/2
0.94
-0.12
-0.48
-0.17
3/4
0.82
-0.16
-0.66
-0.081
1
0.65
-0.18
-0.74
0.022

Therefore, 𝑧(1) ≈ 0.65.
We can also plot the points (𝑥, 𝑦) to get an approximate picture of the curve.
The function eulers_method_2x2_plot will do this; in order to use it, we need to define functions 𝑓 and 𝑔 which takes one
argument with three coordinates: (𝑡, 𝑥, 𝑦).
sage: f = lambda z: z[2]# f(t,x,y) = y
sage: g = lambda z: -sin(z[1]) # g(t,x,y) = -sin(x)sage: P = eulers_method_2x2_plot(f,g, 0.0, 0.75, 0.0, 0.1, 1.0)>>> from sage.all import *>>> f = lambda z: z[Integer(2)]# f(t,x,y) = y>>> g = lambda z: -sin(z[Integer(1)]) # g(t,x,y) = -sin(x)>>> P = eulers_method_2x2_plot(f,g, RealNumber( 0.0 ), RealNumber( 0.75 ), RealNumber(
˓→ 0.0 ), RealNumber( 0.1 ), RealNumber( 1.0 ))At this point, P is storing two plots: P[0], the plot of 𝑥 vs. 𝑡, and P[1], the plot of 𝑦 vs. 𝑡. We can plot both of theseas follows:
sage: show(P[0] + P[1])>>> from sage.all import *>>> show(P[Integer(0)] + P[Integer(1)])(For more on plotting, see Plotting.)

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2.4.5 Special functions
Several orthogonal polynomials and special functions are implemented, using both PARI [GAP] and Maxima [Max].
These are documented in the appropriate sections (“Orthogonal polynomials” and “Special functions”, respectively) of
the Sage reference manual.
sage: x = polygen(QQ, x )sage: chebyshev_U(2,x)
4*x^2 - 1sage: bessel_I(1,1).n(250)
0.56515910399248502720769602760986330732889962162109200948029448947925564096
sage: bessel_I(1,1).n()0.565159103992485
sage: bessel_I(2,1.1).n()0.167089499251049>>> from sage.all import *>>> x = polygen(QQ, x )>>> chebyshev_U(Integer(2),x)4*x^2 - 1>>> bessel_I(Integer(1),Integer(1)).n(Integer(250))
0.56515910399248502720769602760986330732889962162109200948029448947925564096>>> bessel_I(Integer(1),Integer(1)).n()0.565159103992485>>> bessel_I(Integer(2),RealNumber( 1.1 )).n()0.167089499251049

At this point, Sage has only wrapped these functions for numerical use. For symbolic use, please use the Maxima interface
directly, as in the following example:
sage: maxima.eval("f:bessel_y(v, w)")
bessel_y(v,w)
sage: maxima.eval("diff(f,w)")(bessel_y(v-1,w)-bessel_y(v+1,w))/2>>> from sage.all import *>>> maxima.eval("f:bessel_y(v, w)")
bessel_y(v,w)>>> maxima.eval("diff(f,w)")(bessel_y(v-1,w)-bessel_y(v+1,w))/2

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2.4.6 Vector calculus
See the Vector Calculus Tutorial.

2.5 Plotting
Sage can produce two-dimensional and three-dimensional plots.

2.5.1 Two-dimensional Plots
In two dimensions, Sage can draw circles, lines, and polygons; plots of functions in rectangular coordinates; and also polarplots, contour plots and vector field plots. We present examples of some of these here. For more examples of plottingwith Sage, see Solving Differential Equations and Maxima, and also the Sage Constructions documentation.
This command produces a yellow circle of radius 1, centered at the origin:sage: circle((0,0), 1, rgbcolor=(1,1,0))Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),˓→Integer(0)))
Graphics object consisting of 1 graphics primitive

You can also produce a filled circle:
sage: circle((0,0), 1, rgbcolor=(1,1,0), fill=True)
Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),˓→Integer(0)), fill=True)
Graphics object consisting of 1 graphics primitive

You can also create a circle by assigning it to a variable; this does not plot it:
sage: c = circle((0,0), 1, rgbcolor=(1,1,0))>>> from sage.all import *>>> c = circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),˓→Integer(0)))

To plot it, use c.show() or show(c), as follows:
sage: c.show()>>> from sage.all import *>>> c.show()

Alternatively, evaluating c.save( filename.png ) will save the plot to the given file.
Now, these ‘circles’ look more like ellipses because the axes are scaled differently. You can fix this:28

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sage: c.show(aspect_ratio=1)>>> from sage.all import *>>> c.show(aspect_ratio=Integer(1))

The command show(c, aspect_ratio=1) accomplishes the same thing, or you can save the picture using c.
save( filename.png , aspect_ratio=1).
It’s easy to plot basic functions:sage: plot(cos, (-5,5))Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> plot(cos, (-Integer(5),Integer(5)))
Graphics object consisting of 1 graphics primitive

Once you specify a variable name, you can create parametric plots also:
sage: x = var( x )sage: parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> x = var( x )>>> parametric_plot((cos(x),sin(x)**Integer(3)),(x,Integer(0),Integer(2)*pi),
˓→rgbcolor=hue(RealNumber( 0.6 )))Graphics object consisting of 1 graphics primitive

It’s important to notice that the axes of the plots will only intersect if the origin is in the viewing range of the graph, andthat with sufficiently large values scientific notation may be used:
sage: plot(x^2,(x,300,500))
Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> plot(x**Integer(2),(x,Integer(300),Integer(500)))
Graphics object consisting of 1 graphics primitive

You can combine several plots by adding them:sage: x = var( x )sage: p1 = parametric_plot((cos(x),sin(x)),(x,0,2*pi),rgbcolor=hue(0.2))
sage: p2 = parametric_plot((cos(x),sin(x)^2),(x,0,2*pi),rgbcolor=hue(0.4))
sage: p3 = parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
sage: show(p1+p2+p3, axes=false)>>> from sage.all import *>>> x = var( x )>>> p1 = parametric_plot((cos(x),sin(x)),(x,Integer(0),Integer(2)*pi),
˓→rgbcolor=hue(RealNumber( 0.2 )))>>> p2 = parametric_plot((cos(x),sin(x)**Integer(2)),(x,Integer(0),Integer(2)*pi),
˓→rgbcolor=hue(RealNumber( 0.4 )))>>> p3 = parametric_plot((cos(x),sin(x)**Integer(3)),(x,Integer(0),Integer(2)*pi),
˓→rgbcolor=hue(RealNumber( 0.6 )))>>> show(p1+p2+p3, axes=false)

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A good way to produce filled-in shapes is to produce a list of points (L in the example below) and then use the polygoncommand to plot the shape with boundary formed by those points. For example, here is a green deltoid:
sage: L = [[-1+cos(pi*i/100)*(1+cos(pi*i/100)),....:
2*sin(pi*i/100)*(1-cos(pi*i/100))] for i in range(200)]
sage: p = polygon(L, rgbcolor=(1/8,3/4,1/2))
sage: pGraphics object consisting of 1 graphics primitive>>> from sage.all import *>>> L = [[-Integer(1)+cos(pi*i/Integer(100))*(Integer(1)+cos(pi*i/Integer(100))),...
Integer(2)*sin(pi*i/Integer(100))*(Integer(1)-cos(pi*i/Integer(100)))] for i␣˓→in range(Integer(200))]>>> p = polygon(L, rgbcolor=(Integer(1)/Integer(8),Integer(3)/Integer(4),Integer(1)/˓→Integer(2)))>>> p
Graphics object consisting of 1 graphics primitive

Type show(p, axes=false) to see this without any axes.
You can add text to a plot:
sage: L = [[6*cos(pi*i/100)+5*cos((6/2)*pi*i/100),....:
6*sin(pi*i/100)-5*sin((6/2)*pi*i/100)] for i in range(200)]
sage: p = polygon(L, rgbcolor=(1/8,1/4,1/2))
sage: t = text("hypotrochoid", (5,4), rgbcolor=(1,0,0))
sage: show(p+t)>>> from sage.all import *>>> L = [[Integer(6)*cos(pi*i/Integer(100))+Integer(5)*cos((Integer(6)/˓→Integer(2))*pi*i/Integer(100)),...
Integer(6)*sin(pi*i/Integer(100))-Integer(5)*sin((Integer(6)/Integer(2))*pi*i/
˓→Integer(100))] for i in range(Integer(200))]>>> p = polygon(L, rgbcolor=(Integer(1)/Integer(8),Integer(1)/Integer(4),Integer(1)/˓→Integer(2)))>>> t = text("hypotrochoid", (Integer(5),Integer(4)), rgbcolor=(Integer(1),Integer(0),˓→Integer(0)))>>> show(p+t)

Calculus teachers draw the following plot frequently on the board: not just one branch of arcsin but rather several of them:i.e., the plot of 𝑦 = sin(𝑥) for 𝑥 between −2𝜋 and 2𝜋, flipped about the 45 degree line. The following Sage commands
construct this:sage: v = [(sin(x),x) for x in srange(-2*float(pi),2*float(pi),0.1)]
sage: line(v)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> v = [(sin(x),x) for x in srange(-Integer(2)*float(pi),Integer(2)*float(pi),˓→RealNumber( 0.1 ))]>>> line(v)Graphics object consisting of 1 graphics primitive

Since the tangent function has a larger range than sine, if you use the same trick to plot the inverse tangent, you shouldchange the minimum and maximum coordinates for the x-axis:

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sage: v = [(tan(x),x) for x in srange(-2*float(pi),2*float(pi),0.01)]
sage: show(line(v), xmin=-20, xmax=20)>>> from sage.all import *>>> v = [(tan(x),x) for x in srange(-Integer(2)*float(pi),Integer(2)*float(pi),˓→RealNumber( 0.01 ))]>>> show(line(v), xmin=-Integer(20), xmax=Integer(20))

Sage also computes polar plots, contour plots and vector field plots (for special types of functions). Here is an example
of a contour plot:
sage: f = lambda x,y: cos(x*y)
sage: contour_plot(f, (-4, 4), (-4, 4))Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> f = lambda x,y: cos(x*y)>>> contour_plot(f, (-Integer(4), Integer(4)), (-Integer(4), Integer(4)))
Graphics object consisting of 1 graphics primitive

2.5.2 Three-Dimensional Plots
Sage can also be used to create three-dimensional plots. In both the notebook and the REPL, these plots will be displayed
by default using the open source package [ThreeJS], which supports interactively rotating and zooming the figure with themouse.
Use plot3d to graph a function of the form 𝑓 (𝑥, 𝑦) = 𝑧:
sage: x, y = var( x,y )sage: plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))Graphics3d Object>>> from sage.all import *>>> x, y = var( x,y )>>> plot3d(x**Integer(2) + y**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),˓→Integer(2)))
Graphics3d Object

Alternatively, you can use parametric_plot3d to graph a parametric surface where each of 𝑥, 𝑦, 𝑧 is determined bya function of one or two variables (the parameters, typically 𝑢 and 𝑣). The previous plot can be expressed parametricallyas follows:
sage: u, v = var( u, v )sage: f_x(u, v) = u
sage: f_y(u, v) = v
sage: f_z(u, v) = u^2 + v^2
sage: parametric_plot3d([f_x, f_y, f_z], (u, -2, 2), (v, -2, 2))
Graphics3d Object>>> from sage.all import *>>> u, v = var( u, v )>>> __tmp__=var("u,v"); f_x = symbolic_expression(u).function(u,v)>>> __tmp__=var("u,v"); f_y = symbolic_expression(v).function(u,v)>>> __tmp__=var("u,v"); f_z = symbolic_expression(u**Integer(2) + v**Integer(2)).
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(continued from previous page)function(u,v)>>> parametric_plot3d([f_x, f_y, f_z], (u, -Integer(2), Integer(2)), (v, -Integer(2),␣
˓→Integer(2)))
Graphics3d Object
˓→The third way to plot a 3D surface in Sage is implicit_plot3d, which graphs a contour of a function like 𝑓 (𝑥, 𝑦, 𝑧) =0 (this defines a set of points). We graph a sphere using the classical formula:
sage: x, y, z = var( x, y, z )sage: implicit_plot3d(x^2 + y^2 + z^2 - 4, (x,-2, 2), (y,-2, 2), (z,-2, 2))
Graphics3d Object>>> from sage.all import *>>> x, y, z = var( x, y, z )>>> implicit_plot3d(x**Integer(2) + y**Integer(2) + z**Integer(2) - Integer(4), (x,˓→Integer(2), Integer(2)), (y,-Integer(2), Integer(2)), (z,-Integer(2), Integer(2)))
Graphics3d Object

Here are some more examples:
Yellow Whitney’s umbrella:
sage: u, v = var( u,v )sage: fx = u*v
sage: fy = u
sage: fz = v^2
sage: parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -1, 1),....:frame=False, color="yellow")
Graphics3d Object>>> from sage.all import *>>> u, v = var( u,v )>>> fx = u*v>>> fy = u>>> fz = v**Integer(2)>>> parametric_plot3d([fx, fy, fz], (u, -Integer(1), Integer(1)), (v, -Integer(1),␣
˓→Integer(1)),...
frame=False, color="yellow")
Graphics3d Object

Cross cap:
sage: u, v = var( u,v )sage: fx = (1+cos(v))*cos(u)sage: fy = (1+cos(v))*sin(u)sage: fz = -tanh((2/3)*(u-pi))*sin(v)sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),....:
frame=False, color="red")Graphics3d Object>>> from sage.all import *>>> u, v = var( u,v )>>> fx = (Integer(1)+cos(v))*cos(u)>>> fy = (Integer(1)+cos(v))*sin(u)>>> fz = -tanh((Integer(2)/Integer(3))*(u-pi))*sin(v)>>> parametric_plot3d([fx, fy, fz], (u, Integer(0), Integer(2)*pi), (v, Integer(0),␣
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(continued from previous page)

Integer(2)*pi),...
frame=False, color="red")Graphics3d Object
˓→Twisted torus:
sage: u, v = var( u,v )sage: fx = (3+sin(v)+cos(u))*cos(2*v)
sage: fy = (3+sin(v)+cos(u))*sin(2*v)
sage: fz = sin(u)+2*cos(v)sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),....:
frame=False, color="red")Graphics3d Object>>> from sage.all import *>>> u, v = var( u,v )>>> fx = (Integer(3)+sin(v)+cos(u))*cos(Integer(2)*v)>>> fy = (Integer(3)+sin(v)+cos(u))*sin(Integer(2)*v)>>> fz = sin(u)+Integer(2)*cos(v)>>> parametric_plot3d([fx, fy, fz], (u, Integer(0), Integer(2)*pi), (v, Integer(0),␣
˓→Integer(2)*pi),...
frame=False, color="red")Graphics3d Object

Lemniscate:sage: x, y, z = var( x,y,z )sage: f(x, y, z) = 4*x^2 * (x^2 + y^2 + z^2 + z) + y^2 * (y^2 + z^2 - 1)sage: implicit_plot3d(f, (x, -0.5, 0.5), (y, -1, 1), (z, -1, 1))
Graphics3d Object>>> from sage.all import *>>> x, y, z = var( x,y,z )>>> __tmp__=var("x,y,z"); f = symbolic_expression(Integer(4)*x**Integer(2) *␣˓→(x**Integer(2) + y**Integer(2) + z**Integer(2) + z) + y**Integer(2) *␣
˓→(y**Integer(2) + z**Integer(2) - Integer(1))).function(x,y,z)>>> implicit_plot3d(f, (x, -RealNumber( 0.5 ), RealNumber( 0.5 )), (y, -Integer(1),␣
˓→Integer(1)), (z, -Integer(1), Integer(1)))
Graphics3d Object

2.6 Some Common Issues with Functions
Some aspects of defining functions (e.g., for differentiation or plotting) can be confusing. In this section we try to addresssome of the relevant issues.
Here are several ways to define things which might deserve to be called “functions”:
1. Define a Python function, as described in Functions, Indentation, and Counting. These functions can be plotted, but
not differentiated or integrated.
sage: def f(z): return z^2sage: type(f)<... function >sage: f(3)(continues on next page)2.6. Some Common Issues with Functions

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(continued from previous page)9
sage: plot(f, 0, 2)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> def f(z): return z**Integer(2)>>> type(f)<... function >>>> f(Integer(3))
9>>> plot(f, Integer(0), Integer(2))
Graphics object consisting of 1 graphics primitive

In the last line, note the syntax. Using plot(f(z), 0, 2) instead will give a NameError, because z is a dummy
variable in the definition of f and is not defined outside of that definition. In order to be able to use f(z) in the plot
command, z (or whatever is desired) needs to be defined as a variable. We can use the syntax below, or in the next itemin our list.
sage: var( z )# define z to be a variablezsage: f(z)z^2
sage: plot(f(z), 0, 2)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> var( z )# define z to be a variablez>>> f(z)z^2>>> plot(f(z), Integer(0), Integer(2))
Graphics object consisting of 1 graphics primitive

At this point, f(z) is a symbolic expression, the next item in our list.
2. Define a “callable symbolic expression”. These can be plotted, differentiated, and integrated.
sage: g(x) = x^2
sage: g# g sends x to x^2x |--> x^2
sage: g(3)9sage: Dg = g.derivative(); Dg
x |--> 2*x
sage: Dg(3)6sage: type(g)<class sage.symbolic.expression.Expression >sage: plot(g, 0, 2)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> __tmp__=var("x"); g = symbolic_expression(x**Integer(2)).function(x)>>> g# g sends x to x^2x |--> x^2>>> g(Integer(3))(continues on next page)

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(continued from previous page)9>>> Dg = g.derivative(); Dg
x |--> 2*x>>> Dg(Integer(3))
6>>> type(g)<class sage.symbolic.expression.Expression >>>> plot(g, Integer(0), Integer(2))
Graphics object consisting of 1 graphics primitive

Note that while g is a callable symbolic expression, g(x) is a related, but different sort of object, which can also beplotted, differentated, etc., albeit with some issues: see item 5 below for an illustration.
sage: g(x)x^2
sage: type(g(x))<class sage.symbolic.expression.Expression >sage: g(x).derivative()
2*x
sage: plot(g(x), 0, 2)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> g(x)x^2>>> type(g(x))<class sage.symbolic.expression.Expression >>>> g(x).derivative()
2*x>>> plot(g(x), Integer(0), Integer(2))
Graphics object consisting of 1 graphics primitive

3. Use a pre-defined Sage ‘calculus function’. These can be plotted, and with a little help, differentiated, and integrated.
sage: type(sin)<class sage.functions.trig.Function_sin >sage: plot(sin, 0, 2)Graphics object consisting of 1 graphics primitive
sage: type(sin(x))<class sage.symbolic.expression.Expression >sage: plot(sin(x), 0, 2)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> type(sin)<class sage.functions.trig.Function_sin >>>> plot(sin, Integer(0), Integer(2))
Graphics object consisting of 1 graphics primitive>>> type(sin(x))<class sage.symbolic.expression.Expression >>>> plot(sin(x), Integer(0), Integer(2))
Graphics object consisting of 1 graphics primitive

By itself, sin cannot be differentiated, at least not to produce cos.

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sage: f = sin
sage: f.derivative()
Traceback (most recent call last):...
AttributeError: ...
>>> from sage.all import *>>> f = sin>>> f.derivative()
Traceback (most recent call last):...
AttributeError: ...

Using f = sin(x) instead of sin works, but it is probably even better to use f(x) = sin(x) to define a callable
symbolic expression.
sage: S(x) = sin(x)sage: S.derivative()x |--> cos(x)>>> from sage.all import *>>> __tmp__=var("x"); S = symbolic_expression(sin(x)).function(x)>>> S.derivative()x |--> cos(x)Here are some common problems, with explanations:
4. Accidental evaluation.
sage: def h(x):....:if x<2:....:
return 0
....:
else:
....:
return x-2>>> from sage.all import *>>> def h(x):...
if x<Integer(2):...
return Integer(0)
...
else:
...
return x-Integer(2)The issue: plot(h(x), 0, 4) plots the line 𝑦 = 𝑥 − 2, not the multi-line function defined by h. The reason? In thecommand plot(h(x), 0, 4), first h(x) is evaluated: this means plugging the symbolic variable x into the function
h. So, the inequality x < 2 evaluates to False first, and hence h(x) evaluates to x - 2. This can be seen withsage: bool(x < 2)
False
sage: h(x)x - 2>>> from sage.all import *>>> bool(x < Integer(2))
False>>> h(x)x - 2

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Note that here there are two different x: the Python variable used to define the function h (which is local to its definition)and the symbolic variable x which is available on startup in Sage.
The solution: don’t use plot(h(x), 0, 4); instead, use
sage: plot(h, 0, 4)Graphics object consisting of 1 graphics primitive>>> from sage.all import *>>> plot(h, Integer(0), Integer(4))
Graphics object consisting of 1 graphics primitive

5. Accidentally producing a constant instead of a function.
sage: f = x
sage: g = f.derivative()sage: g
1>>> from sage.all import *>>> f = x>>> g = f.derivative()>>> g
1

The problem: g(3), for example, returns an error, saying “ValueError: the number of arguments must be less than orequal to 0.”
sage: type(f)<class sage.symbolic.expression.Expression >sage: type(g)<class sage.symbolic.expression.Expression >>>> from sage.all import *>>> type(f)<class sage.symbolic.expression.Expression >>>> type(g)<class sage.symbolic.expression.Expression >g is not a function, it’s a constant, so it has no variables associated to it, and you can’t plug anything into it.
The solution: there are several options.
• Define f initially to be a symbolic expression.
sage: f(x) = x# instead of f = x
sage: g = f.derivative()sage: gx |--> 1
sage: g(3)1sage: type(g)<class sage.symbolic.expression.Expression >>>> from sage.all import *>>> __tmp__=var("x"); f = symbolic_expression(x˓→x).function(x)# instead off =␣(continues on next page)2.6. Some Common Issues with Functions

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(continued from previous page)>>> g = f.derivative()>>> gx |--> 1>>> g(Integer(3))
1>>> type(g)<class sage.symbolic.expression.Expression >• Or with f as defined originally, define g to be a symbolic expression.
sage: f = x
sage: g(x) = f.derivative() # instead of g = f.derivative()sage: gx |--> 1
sage: g(3)1sage: type(g)<class sage.symbolic.expression.Expression >>>> from sage.all import *>>> f = x>>> __tmp__=var("x"); g = symbolic_expression(f.derivative()
˓→ofg = f.derivative()>>> gx |--> 1>>> g(Integer(3))
1>>> type(g)<class sage.symbolic.expression.Expression >).function(x)# instead␣• Or with f and g as defined originally, specify the variable for which you are substituting.
sage: f = x
sage: g = f.derivative()sage: g1
sage: g(x=3)# instead of
1g(3)>>> from sage.all import *>>> f = x>>> g = f.derivative()>>> g
1>>> g(x=Integer(3))# instead of
1g(3)Finally, here’s one more way to tell the difference between the derivatives of f = x and f(x) = x
sage: f(x) = x
sage: g = f.derivative()sage: g.variables() # the variables present in g()sage: g.arguments() # the arguments which can be plugged into g(x,)
sage: f = x(continues on next page)

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(continued from previous page)

sage: h = f.derivative()sage: h.variables()()
sage: h.arguments()()>>> from sage.all import *>>> __tmp__=var("x"); f = symbolic_expression(x).function(x)>>> g = f.derivative()>>> g.variables() # the variables present in g()>>> g.arguments() # the arguments which can be plugged into g(x,)>>> f = x>>> h = f.derivative()>>> h.variables()()>>> h.arguments()()

As this example has been trying to illustrate, h accepts no arguments, and this is why h(3) returns an error.

2.7 Basic Rings
When defining matrices, vectors, or polynomials, it is sometimes useful and sometimes necessary to specify the “ring”
over which it is defined. A ring is a mathematical construction in which there are well-behaved notions of addition andmultiplication; if you’ve never heard of them before, you probably just need to know about these four commonly usedrings:• the integers {..., −1, 0, 1, 2, ...}, called ZZ in Sage.
• the rational numbers – i.e., fractions, or ratios, of integers – called QQ in Sage.
• the real numbers, called RR in Sage.
• the complex numbers, called CC in Sage.
You may need to know about these distinctions because the same polynomial, for example, can be treateddifferently√
depending on the ring over which it is defined. For instance, the polynomial 𝑥2 − 2 has two roots, ± 2. Those roots
are not rational, so if you are working with polynomials with rational coefficients, the polynomial won’t factor. With realcoefficients, it will. Therefore you may want to specify the ring to insure that you are getting the information you expect.
The following two commands defines the sets of polynomials with rational coefficients and real coefficients, respectively.
The sets are named “ratpoly” and “realpoly”, but these aren’t important here; however, note that the strings “.<t>” and“.<z>” name the variables used in the two cases.
sage: ratpoly.<t> = PolynomialRing(QQ)
sage: realpoly.<z> = PolynomialRing(RR)>>> from sage.all import *>>> ratpoly = PolynomialRing(QQ, names=( t ,)); (t,) = ratpoly._first_ngens(1)>>> realpoly = PolynomialRing(RR, names=( z ,)); (z,) = realpoly._first_ngens(1)Now we illustrate the point about factoring 𝑥2 − 2:

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sage: factor(t^2-2)t^2 - 2sage: factor(z^2-2)(z - 1.41421356237310) * (z + 1.41421356237310)>>> from sage.all import *>>> factor(t**Integer(2)-Integer(2))
t^2 - 2>>> factor(z**Integer(2)-Integer(2))(z - 1.41421356237310) * (z + 1.41421356237310)

Similar comments apply to matrices: the row-reduced form of a matrix can depend on the ring over which it is defined,
as can its eigenvalues and eigenvectors. For more about constructing polynomials, see Polynomials, and for more about
matrices, see Linear Algebra.
The symbol I represents the square root of −1; i is a synonym for I. Of course, this is not a rational number:sage: i # square root of -1
I
sage: i in QQ
False>>> from sage.all import *>>> i # square root of -1
I>>> i in QQ
False

Note: The above code may not work as expected if the variable i has been assigned a different value, for example, if it
was used as a loop variable. If this is the case, typesage: reset( i )>>> from sage.all import *>>> reset( i )to get the original complex value of i.
There is one subtlety in defining complex numbers: as mentioned above, the symbol i represents a square root of −1,
but it is a formal square root of −1 as an algebraic number. Calling CC(i) or CC.0 or CC.gen(0) returns thecomplex square root of −1. Arithmetic involving different kinds of numbers is possible by so-called coercion, see Parents,
Conversion and Coercion.
sage: i = CC(i)# floating point complex number
sage: i == CC.0
True
sage: a, b = 4/3, 2/3
sage: z = a + b*i
sage: z1.33333333333333 + 0.666666666666667*I
sage: z.imag()# imaginary part0.666666666666667
sage: z.real() == a# automatic coercion before comparison
Truesage: a + b2(continues on next page)

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sage: 2*b == aTruesage: parent(2/3)
Rational Field
sage: parent(4/2)
Rational Field
sage: 2/3 + 0.1
0.766666666666667
sage: 0.1 + 2/3
0.766666666666667# automatic coercion before addition# coercion rules are symmetric in Sage>>> from sage.all import *>>> i = CC(i)# floating point complex number>>> i == CC.gen(0)True>>> a, b = Integer(4)/Integer(3), Integer(2)/Integer(3)>>> z = a + b*i>>> z
1.33333333333333 + 0.666666666666667*I>>> z.imag()# imaginary part0.666666666666667>>> z.real() == a# automatic coercion before comparison
True>>> a + b
2>>> Integer(2)*b == aTrue>>> parent(Integer(2)/Integer(3))
Rational Field>>> parent(Integer(4)/Integer(2))
Rational Field>>> Integer(2)/Integer(3) + RealNumber( 0.1 )# automatic coercion before␣
˓→addition
0.766666666666667>>> RealNumber( 0.1 ) + Integer(2)/Integer(3)# coercion rules are symmetric in␣˓→Sage0.766666666666667

Here are more examples of basic rings in Sage. As noted above, the ring of rational numbers may be referred to using QQ,
or also RationalField() (a field is a ring in which the multiplication is commutative and in which every nonzero
element has a reciprocal in that ring, so the rationals form a field, but the integers don’t):
sage: RationalField()
Rational Field
sage: QQ
Rational Field
sage: 1/2 in QQ
True>>> from sage.all import *>>> RationalField()
Rational Field>>> QQ
Rational Field>>> Integer(1)/Integer(2) in QQ
True

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The decimal number 1.2 is considered to be in QQ: decimal numbers which happen to√also be rational can be “coerced”
into the rational numbers (see Parents, Conversion and Coercion). The numbers 𝜋 and 2 are not rational, though:sage: 1.2 in QQ
True
sage: pi in QQ
False
sage: pi in RR
True
sage: sqrt(2) in QQ
False
sage: sqrt(2) in CC
True>>> from sage.all import *>>> RealNumber( 1.2 ) in QQ
True>>> pi in QQ
False>>> pi in RR
True>>> sqrt(Integer(2)) in QQ
False>>> sqrt(Integer(2)) in CC
True

For use in higher mathematics, Sage also knows about other rings, such as finite fields, 𝑝-adic integers, the ring of algebraic
numbers, polynomial rings, and matrix rings. Here are constructions of some of these:
sage: GF(3)Finite Field of size 3
sage: GF(27, a ) # need to name the generator if not a prime field
Finite Field in a of size 3^3
sage: Zp(5)5-adic Ring with capped relative precision 20
sage: sqrt(3) in QQbar # algebraic closure of QQ
True>>> from sage.all import *>>> GF(Integer(3))
Finite Field of size 3>>> GF(Integer(27), a ) # need to name the generator if not a prime field
Finite Field in a of size 3^3>>> Zp(Integer(5))
5-adic Ring with capped relative precision 20>>> sqrt(Integer(3)) in QQbar # algebraic closure of QQ
True

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2.8 Linear Algebra
Sage provides standard constructions from linear algebra, e.g., the characteristic polynomial, echelon form, trace, decomposition, etc., of a matrix.
Creation of matrices and matrix multiplication is easy and natural:
sage: A = Matrix([[1,2,3],[3,2,1],[1,1,1]])
sage: w = vector([1,1,-4])
sage: w*A(0, 0, 0)
sage: A*w(-9, 1, -2)
sage: kernel(A)Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:[ 1 1 -4]>>> from sage.all import *>>> A = Matrix([[Integer(1),Integer(2),Integer(3)],[Integer(3),Integer(2),Integer(1)],
˓→[Integer(1),Integer(1),Integer(1)]])>>> w = vector([Integer(1),Integer(1),-Integer(4)])>>> w*A(0, 0, 0)>>> A*w(-9, 1, -2)>>> kernel(A)Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:[ 1 1 -4]

Note that in Sage, the kernel of a matrix 𝐴 is the “left kernel”, i.e. the space of vectors 𝑤 such that 𝑤𝐴 = 0.
Solving matrix equations is easy, using the method solve_right. Evaluating A.solve_right(Y) returns a matrix(or vector) 𝑋 so that 𝐴𝑋 = 𝑌 :sage: Y = vector([0, -4, -1])
sage: X = A.solve_right(Y)sage: X(-2, 1, 0)
sage: A * X# checking our answer...
(0, -4, -1)>>> from sage.all import *>>> Y = vector([Integer(0), -Integer(4), -Integer(1)])>>> X = A.solve_right(Y)>>> X(-2, 1, 0)>>> A * X
# checking our answer...
(0, -4, -1)

If there is no solution, Sage returns an error:
sage: A.solve_right(w)Traceback (most recent call last):...
ValueError: matrix equation has no solutions

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Tutorial, Release 10.4>>> from sage.all import *>>> A.solve_right(w)Traceback (most recent call last):...
ValueError: matrix equation has no solutions

Similarly, use A.solve_left(Y) to solve for 𝑋 in 𝑋𝐴 = 𝑌 .
Sage can also compute eigenvalues and eigenvectors:
sage: A = matrix([[0, 4], [-1, 0]])sage: A.eigenvalues ()[-2*I, 2*I]
sage: B = matrix([[1, 3], [3, 1]])
sage: B.eigenvectors_left()[(4, [(1, 1)], 1), (-2, [(1, -1)], 1)]>>> from sage.all import *>>> A = matrix([[Integer(0), Integer(4)], [-Integer(1), Integer(0)]])>>> A.eigenvalues ()[-2*I, 2*I]>>> B = matrix([[Integer(1), Integer(3)], [Integer(3), Integer(1)]])>>> B.eigenvectors_left()[(4, [(1, 1)], 1), (-2, [(1, -1)], 1)](The syntax for the output of eigenvectors_left is a list of triples: (eigenvalue, eigenvector, multiplicity).) Eigenvalues and eigenvectors over QQ or RR can also be computed using Maxima (see Maxima below).
As noted in Basic Rings, the ring over which a matrix is defined affects some of its properties. In the following, the firstargument to the matrix command tells Sage to view the matrix as a matrix of integers (the ZZ case), a matrix of rational
numbers (QQ), or a matrix of reals (RR):
sage: AZ = matrix(ZZ, [[2,0], [0,1]])
sage: AQ = matrix(QQ, [[2,0], [0,1]])
sage: AR = matrix(RR, [[2,0], [0,1]])
sage: AZ.echelon_form()
[2 0]
[0 1]
sage: AQ.echelon_form()
[1 0]
[0 1]
sage: AR.echelon_form()[ 1.00000000000000 0.000000000000000][0.000000000000000 1.00000000000000]>>> from sage.all import *>>> AZ = matrix(ZZ, [[Integer(2),Integer(0)], [Integer(0),Integer(1)]])>>> AQ = matrix(QQ, [[Integer(2),Integer(0)], [Integer(0),Integer(1)]])>>> AR = matrix(RR, [[Integer(2),Integer(0)], [Integer(0),Integer(1)]])(continues on next page)

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(continued from previous page)>>> AZ.echelon_form()
[2 0]
[0 1]
>>> AQ.echelon_form()
[1 0]
[0 1]
>>> AR.echelon_form()[ 1.00000000000000 0.000000000000000][0.000000000000000 1.00000000000000]

For computing eigenvalues and eigenvectors of matrices over floating point real or complex numbers, the matrix should be
defined over RDF (Real Double Field) or CDF (Complex Double Field), respectively. If no ring is specified and floating
point real or complex numbers are used then by default the matrix is defined over the RR or CC fields, respectively, which
do not support these computations for all the cases:
sage: ARDF = matrix(RDF, [[1.2, 2], [2, 3]])sage: ARDF.eigenvalues() # rel tol 8e-16[-0.09317121994613098, 4.293171219946131]
sage: ACDF = matrix(CDF, [[1.2, I], [2, 3]])sage: ACDF.eigenvectors_right() # rel tol 3e-15[(0.8818456983293743 - 0.8209140653434135*I, [(0.7505608183809549, -0.616145932704589␣
˓→+ 0.2387941530333261*I)], 1),(3.3181543016706256 + 0.8209140653434133*I, [(0.14559469829270957 + 0.
˓→3756690858502104*I, 0.9152458258662108)], 1)]>>> from sage.all import *>>> ARDF = matrix(RDF, [[RealNumber( 1.2 ), Integer(2)], [Integer(2), Integer(3)]])>>> ARDF.eigenvalues() # rel tol 8e-16[-0.09317121994613098, 4.293171219946131]>>> ACDF = matrix(CDF, [[RealNumber( 1.2 ), I], [Integer(2), Integer(3)]])>>> ACDF.eigenvectors_right() # rel tol 3e-15[(0.8818456983293743 - 0.8209140653434135*I, [(0.7505608183809549, -0.616145932704589␣
˓→+ 0.2387941530333261*I)], 1),(3.3181543016706256 + 0.8209140653434133*I, [(0.14559469829270957 + 0.
˓→3756690858502104*I, 0.9152458258662108)], 1)]

2.8.1 Matrix spaces
We create the space Mat3×3 ( ) of 3 × 3 matrices with rational entries:
sage: M = MatrixSpace(QQ,3)
sage: M
Full MatrixSpace of 3 by 3 dense matrices over Rational Field>>> from sage.all import *>>> M = MatrixSpace(QQ,Integer(3))>>> M
Full MatrixSpace of 3 by 3 dense matrices over Rational Field(To specify the space of 3 by 4 matrices, you would use MatrixSpace(QQ,3,4). If the number of columns isomitted, it defaults to the number of rows, so MatrixSpace(QQ,3) is a synonym for MatrixSpace(QQ,3,3).)The space of matrices is equipped with its canonical basis:

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sage: B = M.basis()
sage: len(B)9sage: B[0,1][0 1 0][0 0 0][0 0 0]>>> from sage.all import *>>> B = M.basis()>>> len(B)9>>> B[Integer(0),Integer(1)][0 1 0][0 0 0][0 0 0]We create a matrix as an element of M.
sage: A = M(range(9)); A[0 1 2][3 4 5][6 7 8]>>> from sage.all import *>>> A = M(range(Integer(9))); A[0 1 2][3 4 5][6 7 8]

Next we compute its reduced row echelon form and kernel.
sage: A.echelon_form()[ 1 0 -1]
[ 0 1 2][ 0 0 0]sage: A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:[ 1 -2 1]>>> from sage.all import *>>> A.echelon_form()[ 1 0 -1]
[ 0 1 2][ 0 0 0]>>> A.kernel()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:[ 1 -2 1]

Next we illustrate computation of matrices defined over finite fields:
sage: M = MatrixSpace(GF(2),4,8)
sage: A = M([1,1,0,0, 1,1,1,1, 0,1,0,0, 1,0,1,1,....:
0,0,1,0, 1,1,0,1, 0,0,1,1, 1,1,1,0])(continues on next page)46

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sage: A
[1 1 0 0 1 1 1 1]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 1 1 1 1 1 0]
sage: rows = A.rows()
sage: A.columns()[(1, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1),(1, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)]
sage: rows[(1, 1, 0, 0, 1, 1, 1, 1), (0, 1, 0, 0, 1, 0, 1, 1),(0, 0, 1, 0, 1, 1, 0, 1), (0, 0, 1, 1, 1, 1, 1, 0)]>>> from sage.all import *>>> M = MatrixSpace(GF(Integer(2)),Integer(4),Integer(8))>>> A = M([Integer(1),Integer(1),Integer(0),Integer(0), Integer(1),Integer(1),˓→Integer(1),Integer(1), Integer(0),Integer(1),Integer(0),Integer(0), Integer(1),˓→Integer(0),Integer(1),Integer(1),...
Integer(0),Integer(0),Integer(1),Integer(0), Integer(1),Integer(1),˓→Integer(0),Integer(1), Integer(0),Integer(0),Integer(1),Integer(1), Integer(1),˓→Integer(1),Integer(1),Integer(0)])>>> A[1 1 0 0 1 1 1 1]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 1 1 1 1 1 0]>>> rows = A.rows()>>> A.columns()[(1, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1),(1, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)]>>> rows[(1, 1, 0, 0, 1, 1, 1, 1), (0, 1, 0, 0, 1, 0, 1, 1),(0, 0, 1, 0, 1, 1, 0, 1), (0, 0, 1, 1, 1, 1, 1, 0)]We make the subspace over F2 spanned by the above rows.
sage: V = VectorSpace(GF(2),8)
sage: S = V.subspace(rows)
sage: S
Vector space of degree 8 and dimension 4 over Finite Field of size 2
Basis matrix:[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]
sage: A.echelon_form()[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]>>> from sage.all import *>>> V = VectorSpace(GF(Integer(2)),Integer(8))>>> S = V.subspace(rows)>>> S
Vector space of degree 8 and dimension 4 over Finite Field of size 2(continues on next page)2.8. Linear Algebra

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Basis matrix:[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]>>> A.echelon_form()[1 0 0 0 0 1 0 0]
[0 1 0 0 1 0 1 1]
[0 0 1 0 1 1 0 1]
[0 0 0 1 0 0 1 1]

The basis of 𝑆 used by Sage is obtained from the non-zero rows of the reduced row echelon form of the matrix ofgenerators of 𝑆.

2.8.2 Sparse Linear Algebra
Sage has support for sparse linear algebra over PIDs.
sage: M = MatrixSpace(QQ, 100, sparse=True)
sage: A = M.random_element(density = 0.05)
sage: E = A.echelon_form()>>> from sage.all import *>>> M = MatrixSpace(QQ, Integer(100), sparse=True)>>> A = M.random_element(density = RealNumber( 0.05 ))>>> E = A.echelon_form()

The multi-modular algorithm in Sage is good for square matrices (but not so good for non-square matrices):sage: M = MatrixSpace(QQ, 50, 100, sparse=True)
sage: A = M.random_element(density = 0.05)
sage: E = A.echelon_form()
sage: M = MatrixSpace(GF(2), 20, 40, sparse=True)
sage: A = M.random_element()
sage: E = A.echelon_form()>>> from sage.all import *>>> M = MatrixSpace(QQ, Integer(50), Integer(100), sparse=True)>>> A = M.random_element(density = RealNumber( 0.05 ))>>> E = A.echelon_form()>>> M = MatrixSpace(GF(Integer(2)), Integer(20), Integer(40), sparse=True)>>> A = M.random_element()>>> E = A.echelon_form()

Note that Python is case sensitive:
sage: M = MatrixSpace(QQ, 10,10, Sparse=True)
Traceback (most recent call last):...
TypeError: ...__init__() got an unexpected keyword argument

Sparse>>> from sage.all import *>>> M = MatrixSpace(QQ, Integer(10),Integer(10), Sparse=True)
Traceback (most recent call last):(continues on next page)48

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(continued from previous page)...
TypeError: ...__init__() got an unexpected keyword argument

Sparse

2.9 Polynomials
In this section we illustrate how to create and use polynomials in Sage.

2.9.1 Univariate Polynomials
There are three ways to create polynomial rings.
sage: R = PolynomialRing(QQ, t )sage: R
Univariate Polynomial Ring in t over Rational Field>>> from sage.all import *>>> R = PolynomialRing(QQ, t )>>> R
Univariate Polynomial Ring in t over Rational Field

This creates a polynomial ring and tells Sage to use (the string) ‘t’ as the indeterminate when printing to the screen.
However, this does not define the symbol t for use in Sage, so you cannot use it to enter a polynomial (such as 𝑡2 + 1)
belonging to R.
An alternate way issage: S = QQ[ t ]sage: S == R
True>>> from sage.all import *>>> S = QQ[ t ]>>> S == R
TrueThis has the same issue regarding t.
A third very convenient way issage: R.<t> = PolynomialRing(QQ)>>> from sage.all import *>>> R = PolynomialRing(QQ, names=( t ,)); (t,) = R._first_ngens(1)orsage: R.<t> = QQ[ t ]>>> from sage.all import *>>> R = QQ[ t ]; (t,) = R._first_ngens(1)or even
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sage: R.<t> = QQ[]>>> from sage.all import *>>> R = QQ[ t ]; (t,) = R._first_ngens(1)This has the additional side effect that it defines the variable t to be the indeterminate of the polynomial ring, so you caneasily construct elements of R, as follows. (Note that the third way is very similar to the constructor notation in Magma,
and just as in Magma it can be used for a wide range of objects.)
sage: poly = (t+1) * (t+2); polyt^2 + 3*t + 2
sage: poly in R
True>>> from sage.all import *>>> poly = (t+Integer(1)) * (t+Integer(2)); polyt^2 + 3*t + 2>>> poly in R
True

Whatever method you use to define a polynomial ring, you can recover the indeterminate as the 0𝑡ℎ generator:
sage: R = PolynomialRing(QQ,
sage: t = R.0
sage: t in R
True>>> from sage.all import *>>> R = PolynomialRing(QQ,>>> t = R.gen(0)>>> t in R
True

t )

t )

Note that a similar construction works with the complex numbers: the complex numbers can be viewed as being generated
over the real numbers by the symbol i; thus we have the following:
sage: CC
Complex Field with 53 bits of precision
sage: CC.0 # 0th generator of CC
1.00000000000000*I>>> from sage.all import *>>> CC
Complex Field with 53 bits of precision>>> CC.gen(0) # 0th generator of CC
1.00000000000000*I

For polynomial rings, you can obtain both the ring and its generator, or just the generator, during ring creation as follows:
sage: R, t = QQ[ t ].objgen()sage: t= QQ[ t ].gen()sage: R, t = objgen(QQ[ t ])sage: t= gen(QQ[ t ])50

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Tutorial, Release 10.4>>> from sage.all import *>>> R, t = QQ[ t ].objgen()>>> t= QQ[ t ].gen()>>> R, t = objgen(QQ[ t ])>>> t= gen(QQ[ t ])Finally we do some arithmetic in[𝑡].

sage: R, t = QQ[ t ].objgen()sage: f = 2*t^7 + 3*t^2 - 15/19
sage: f^2
4*t^14 + 12*t^9 - 60/19*t^7 + 9*t^4 - 90/19*t^2 + 225/361
sage: cyclo = R.cyclotomic_polynomial(7); cyclot^6 + t^5 + t^4 + t^3 + t^2 + t + 1
sage: g = 7 * cyclo * t^5 * (t^5 + 10*t + 2)
sage: g
7*t^16 + 7*t^15 + 7*t^14 + 7*t^13 + 77*t^12 + 91*t^11 + 91*t^10 + 84*t^9+ 84*t^8 + 84*t^7 + 84*t^6 + 14*t^5sage: F = factor(g); F(7) * t^5 * (t^5 + 10*t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)
sage: F.unit()
7
sage: list(F)[(t, 5), (t^5 + 10*t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]>>> from sage.all import *>>> R, t = QQ[ t ].objgen()>>> f = Integer(2)*t**Integer(7) + Integer(3)*t**Integer(2) - Integer(15)/Integer(19)>>> f**Integer(2)4*t^14 + 12*t^9 - 60/19*t^7 + 9*t^4 - 90/19*t^2 + 225/361>>> cyclo = R.cyclotomic_polynomial(Integer(7)); cyclot^6 + t^5 + t^4 + t^3 + t^2 + t + 1>>> g = Integer(7) * cyclo * t**Integer(5) * (t**Integer(5) + Integer(10)*t +␣˓→Integer(2))>>> g
7*t^16 + 7*t^15 + 7*t^14 + 7*t^13 + 77*t^12 + 91*t^11 + 91*t^10 + 84*t^9+ 84*t^8 + 84*t^7 + 84*t^6 + 14*t^5>>> F = factor(g); F(7) * t^5 * (t^5 + 10*t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)>>> F.unit()
7>>> list(F)[(t, 5), (t^5 + 10*t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]

Notice that the factorization correctly takes into account and records the unit part.
If you were to use, e.g., the R.cyclotomic_polynomial function a lot for some research project, in addition
to citing Sage you should make an attempt to find out what component of Sage is being used to actually compute thecyclotomic polynomial and cite that as well. In this case, if you type R.cyclotomic_polynomial?? to see thesource code, you’ll quickly see a line f = pari.polcyclo(n) which means that PARI is being used for computationof the cyclotomic polynomial. Cite PARI in your work as well.
Dividing two polynomials constructs an element of the fraction field (which Sage creates automatically).
sage: x = QQ[ x ].0
sage: f = x^3 + 1; g = x^2 - 17sage: h = f/g; h(continues on next page)2.9. Polynomials

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(continued from previous page)(x^3 + 1)/(x^2 - 17)sage: h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field>>> from sage.all import *>>> x = QQ[ x ].gen(0)>>> f = x**Integer(3) + Integer(1); g = x**Integer(2) - Integer(17)>>> h = f/g; h(x^3 + 1)/(x^2 - 17)>>> h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field

Using Laurent series, one can compute series expansions in the fraction field of QQ[x]:sage: R.<x> = LaurentSeriesRing(QQ); R
Laurent Series Ring in x over Rational Field
sage: 1/(1-x) + O(x^10)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)>>> from sage.all import *>>> R = LaurentSeriesRing(QQ, names=( x ,)); (x,) = R._first_ngens(1); R
Laurent Series Ring in x over Rational Field>>> Integer(1)/(Integer(1)-x) + O(x**Integer(10))
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)

If we name the variable differently, we obtain a different univariate polynomial ring.
sage: R.<x> = PolynomialRing(QQ)
sage: S.<y> = PolynomialRing(QQ)
sage: x == yFalse
sage: R == S
False
sage: R(y)xsage: R(y^2 - 17)x^2 - 17>>> from sage.all import *>>> R = PolynomialRing(QQ, names=( x ,)); (x,) = R._first_ngens(1)>>> S = PolynomialRing(QQ, names=( y ,)); (y,) = S._first_ngens(1)>>> x == yFalse>>> R == S
False>>> R(y)x>>> R(y**Integer(2) - Integer(17))
x^2 - 17

The ring is determined by the variable. Note that making another ring with variable called x does not return a different
ring.
sage: R = PolynomialRing(QQ, "x")sage: T = PolynomialRing(QQ, "x")(continues on next page)

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sage: R == T
True
sage: R is T
True
sage: R.0 == T.0
True>>> from sage.all import *>>> R = PolynomialRing(QQ, "x")>>> T = PolynomialRing(QQ, "x")>>> R == T
True>>> R is T
True>>> R.gen(0) == T.gen(0)True

Sage also has support for power series and Laurent series rings over any base ring. In the following example, we create
an element of F7 [[𝑇 ]] and divide to create an element of F7 ((𝑇 )).
sage: R.<T> = PowerSeriesRing(GF(7)); R
Power Series Ring in T over Finite Field of size 7
sage: f = T + 3*T^2 + T^3 + O(T^4)
sage: f^3T^3 + 2*T^4 + 2*T^5 + O(T^6)
sage: 1/f
T^-1 + 4 + T + O(T^2)
sage: parent(1/f)
Laurent Series Ring in T over Finite Field of size 7>>> from sage.all import *>>> R = PowerSeriesRing(GF(Integer(7)), names=( T ,)); (T,) = R._first_ngens(1); R
Power Series Ring in T over Finite Field of size 7>>> f = T + Integer(3)*T**Integer(2) + T**Integer(3) + O(T**Integer(4))>>> f**Integer(3)T^3 + 2*T^4 + 2*T^5 + O(T^6)>>> Integer(1)/fT^-1 + 4 + T + O(T^2)>>> parent(Integer(1)/f)
Laurent Series Ring in T over Finite Field of size 7

You can also create power series rings using a double-brackets shorthand:sage: GF(7)[[ T ]]Power Series Ring in T over Finite Field of size 7>>> from sage.all import *>>> GF(Integer(7))[[ T ]]Power Series Ring in T over Finite Field of size 7

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2.9.2 Multivariate Polynomials
To work with polynomials of several variables, we declare the polynomial ring and variables first.
sage: R = PolynomialRing(GF(5),3,"z") # here, 3 = number of variables
sage: R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5>>> from sage.all import *>>> R = PolynomialRing(GF(Integer(5)),Integer(3),"z") # here, 3 = number of variables>>> R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5

Just as for defining univariate polynomial rings, there are alternative ways:
sage: GF(5)[ z0, z1, z2 ]Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
sage: R.<z0,z1,z2> = GF(5)[]; RMultivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5>>> from sage.all import *>>> GF(Integer(5))[ z0, z1, z2 ]Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5>>> R = GF(Integer(5))[ z0, z1, z2 ]; (z0, z1, z2,) = R._first_ngens(3); R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5

Also, if you want the variable names to be single letters then you can use the following shorthand:sage: PolynomialRing(GF(5), 3, xyz )Multivariate Polynomial Ring in x, y, z over Finite Field of size 5>>> from sage.all import *>>> PolynomialRing(GF(Integer(5)), Integer(3), xyz )Multivariate Polynomial Ring in x, y, z over Finite Field of size 5

Next let’s do some arithmetic.
sage: z = GF(5)[ z0, z1, z2 ].gens()
sage: z(z0, z1, z2)
sage: (z[0]+z[1]+z[2])^2
z0^2 + 2*z0*z1 + z1^2 + 2*z0*z2 + 2*z1*z2 + z2^2>>> from sage.all import *>>> z = GF(Integer(5))[ z0, z1, z2 ].gens()>>> z(z0, z1, z2)>>> (z[Integer(0)]+z[Integer(1)]+z[Integer(2)])**Integer(2)z0^2 + 2*z0*z1 + z1^2 + 2*z0*z2 + 2*z1*z2 + z2^2

You can also use more mathematical notation to construct a polynomial ring.
sage: R = GF(5)[ x,y,z ]sage: x,y,z = R.gens()
sage: QQ[ x ]Univariate Polynomial Ring in x over Rational Field(continues on next page)

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sage: QQ[ x,y ].gens()(x, y)
sage: QQ[ x ].objgens()(Univariate Polynomial Ring in x over Rational Field, (x,))>>> from sage.all import *>>> R = GF(Integer(5))[ x,y,z ]>>> x,y,z = R.gens()>>> QQ[ x ]Univariate Polynomial Ring in x over Rational Field>>> QQ[ x,y ].gens()(x, y)>>> QQ[ x ].objgens()(Univariate Polynomial Ring in x over Rational Field, (x,))

Multivariate polynomials are implemented in Sage using Python dictionaries and the “distributive representation” of apolynomial. Sage makes some use of Singular [Si], e.g., for computation of gcd’s and Gröbner basis of ideals.
sage: R, (x, y) = PolynomialRing(RationalField(), 2,
sage: f = (x^3 + 2*y^2*x)^2
sage: g = x^2*y^2
sage: f.gcd(g)x^2xy ).objgens()>>> from sage.all import *>>> R, (x, y) = PolynomialRing(RationalField(), Integer(2), xy ).objgens()>>> f = (x**Integer(3) + Integer(2)*y**Integer(2)*x)**Integer(2)>>> g = x**Integer(2)*y**Integer(2)>>> f.gcd(g)x^2

Next we create the ideal (𝑓, 𝑔) generated by 𝑓 and 𝑔, by simply multiplying (f,g) by R (we could also write ideal([f,g]) or ideal(f,g)).
sage: I = (f, g)*R; I
Ideal (x^6 + 4*x^4*y^2 + 4*x^2*y^4, x^2*y^2) of Multivariate Polynomial
Ring in x, y over Rational Field
sage: B = I.groebner_basis(); B[x^6, x^2*y^2]
sage: x^2 in I
False>>> from sage.all import *>>> I = (f, g)*R; I
Ideal (x^6 + 4*x^4*y^2 + 4*x^2*y^4, x^2*y^2) of Multivariate Polynomial
Ring in x, y over Rational Field>>> B = I.groebner_basis(); B[x^6, x^2*y^2]>>> x**Integer(2) in I
False

Incidentally, the Gröbner basis above is not a list but an immutable sequence. This means that it has a universe, parent,
and cannot be changed (which is good because changing the basis would break other routines that use the Gröbner basis).

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sage: B.universe()
Multivariate Polynomial Ring in x, y over Rational Field
sage: B[1] = x
Traceback (most recent call last):...
ValueError: object is immutable; please change a copy instead.
>>> from sage.all import *>>> B.universe()
Multivariate Polynomial Ring in x, y over Rational Field>>> B[Integer(1)] = x
Traceback (most recent call last):...
ValueError: object is immutable; please change a copy instead.

Some (read: not as much as we would like) commutative algebra is available in Sage, implemented via Singular. For
example, we can compute the primary decomposition and associated primes of 𝐼:
sage: I.primary_decomposition()[Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field]
sage: I.associated_primes()[Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]>>> from sage.all import *>>> I.primary_decomposition()[Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field]>>> I.associated_primes()[Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]

2.10 Parents, Conversion and Coercion
This section may seem more technical than the previous, but we believe that it is important to understand the meaning ofparents and coercion in order to use rings and other algebraic structures in Sage effectively and efficiently.
Note that we try to explain notions, but we do not show here how to implement them. An implementation-oriented tutorial
is available as a Sage thematic tutorial.

2.10.1 Elements
If one wants to implement a ring in Python, a first approximation is to create a class for the elements X of that ring andprovide it with the required double underscore methods such as __add__, __sub__, __mul__, of course making
sure that the ring axioms hold.
As Python is a strongly typed (yet dynamically typed) language, one might, at least at first, expect that one implements
one Python class for each ring. After all, Python contains one type <int> for the integers, one type <float> for thereals, and so on. But that approach must soon fail: There are infinitely many rings, and one can not implement infinitely
many classes.
Instead, one may create a hierarchy of classes designed to implement elements of ubiquitous algebraic structures, such asgroups, rings, skew fields, commutative rings, fields, algebras, and so on.
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But that means that elements of fairly different rings can have the same type.
sage: P.<x,y> = GF(3)[]sage: Q.<a,b> = GF(4, z )[]sage: type(x)==type(a)True>>> from sage.all import *>>> P = GF(Integer(3))[ x, y ]; (x, y,) = P._first_ngens(2)>>> Q = GF(Integer(4), z )[ a, b ]; (a, b,) = Q._first_ngens(2)>>> type(x)==type(a)True

On the other hand, one could also have different Python classes providing different implementations of the same mathematical structure (e.g., dense matrices versus sparse matrices)sage: P.<a> = PolynomialRing(ZZ)
sage: Q.<b> = PolynomialRing(ZZ, sparse=True)
sage: R.<c> = PolynomialRing(ZZ, implementation= NTL )sage: type(a); type(b); type(c)<class sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_
˓→flint ><class sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_
˓→category.element_class ><class sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_
˓→ntl >>>> from sage.all import *>>> P = PolynomialRing(ZZ, names=( a ,)); (a,) = P._first_ngens(1)>>> Q = PolynomialRing(ZZ, sparse=True, names=( b ,)); (b,) = Q._first_ngens(1)>>> R = PolynomialRing(ZZ, implementation= NTL , names=( c ,)); (c,) = R._first_
˓→ngens(1)>>> type(a); type(b); type(c)<class sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_
˓→flint ><class sage.rings.polynomial.polynomial_ring.PolynomialRing_integral_domain_with_
˓→category.element_class ><class sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_
˓→ntl >

That poses two problems: On the one hand, if one has elements that are two instances of the same class, then one may
expect that their __add__ method will allow to add them; but one does not want that, if the elements belong to very
different rings. On the other hand, if one has elements belonging to different implementations of the same ring, then one
wants to add them, but that is not straight forward if they belong to different Python classes.
The solution to these problems is called “coercion” and will be explained below.
However, it is essential that each element knows what it is element of. That is available by the method parent():
sage: a.parent(); b.parent(); c.parent()
Univariate Polynomial Ring in a over Integer Ring
Sparse Univariate Polynomial Ring in b over Integer Ring
Univariate Polynomial Ring in c over Integer Ring (using NTL)>>> from sage.all import *>>> a.parent(); b.parent(); c.parent()
Univariate Polynomial Ring in a over Integer Ring
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Sparse Univariate Polynomial Ring in b over Integer Ring
Univariate Polynomial Ring in c over Integer Ring (using NTL)

2.10.2 Parents and categories
Similar to the hierarchy of Python classes addressed to elements of algebraic structures, Sage also provides classes for the
algebraic structures that contain these elements. Structures containing elements are called “parent structures” in Sage, andthere is a base class for them. Roughly parallel to the hierarchy of mathematical notions, one has a hierarchy of classes,
namely for sets, rings, fields, and so on:
sage: isinstance(QQ,Field)
True
sage: isinstance(QQ, Ring)
True
sage: isinstance(ZZ,Field)
False
sage: isinstance(ZZ, Ring)
True>>> from sage.all import *>>> isinstance(QQ,Field)
True>>> isinstance(QQ, Ring)
True>>> isinstance(ZZ,Field)
False>>> isinstance(ZZ, Ring)
True

In algebra, objects sharing the same kind of algebraic structures are collected in so-called “categories”. So, there is a roughanalogy between the class hierarchy in Sage and the hierarchy of categories. However, this analogy of Python classes andcategories shouldn’t be stressed too much. After all, mathematical categories are implemented in Sage as well:sage: Rings()
Category of rings
sage: ZZ.category()
Join of Category of Dedekind domains
and Category of euclidean domains
and Category of noetherian rings
and Category of infinite enumerated sets
and Category of metric spaces
sage: ZZ.category().is_subcategory(Rings())
True
sage: ZZ in Rings()
True
sage: ZZ in Fields()
False
sage: QQ in Fields()
True>>> from sage.all import *>>> Rings()
Category of rings>>> ZZ.category()(continues on next page)

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Join of Category of Dedekind domains
and Category of euclidean domains
and Category of noetherian rings
and Category of infinite enumerated sets
and Category of metric spaces>>> ZZ.category().is_subcategory(Rings())
True>>> ZZ in Rings()
True>>> ZZ in Fields()
False>>> QQ in Fields()
True

While Sage’s class hierarchy is centered at implementation details, Sage’s category framework is more centered on mathematical structure. It is possible to implement generic methods and tests independent of a specific implementation in the
categories.
Parent structures in Sage are supposed to be unique Python objects. For example, once a polynomial ring over a certain
base ring and with a certain list of generators is created, the result is cached:sage: RR[ x , y ] is RR[ x , y ]True>>> from sage.all import *>>> RR[ x , y ] is RR[ x , y ]True

2.10.3 Types versus parents
The type RingElement does not correspond perfectly to the mathematical notion of a ring element. For example,
although square matrices belong to a ring, they are not instances of RingElement:
sage: M = Matrix(ZZ,2,2); M
[0 0]
[0 0]
sage: isinstance(M, RingElement)
False>>> from sage.all import *>>> M = Matrix(ZZ,Integer(2),Integer(2)); M
[0 0]
[0 0]
>>> isinstance(M, RingElement)
False

While parents are unique, equal elements of a parent in Sage are not necessarily identical. This is in contrast to the
behaviour of Python for some (albeit not all) integers:
sage: int(1) is int(1) # Python int
True
sage: int(-15) is int(-15)
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sage: 1 is 1
False

# Sage Integer>>> from sage.all import *>>> int(Integer(1)) is int(Integer(1)) # Python int
True>>> int(-Integer(15)) is int(-Integer(15))
False>>> Integer(1) is Integer(1)# Sage Integer
False

It is important to observe that elements of different rings are in general not distinguished by their type, but by their parent:
sage: a = GF(2)(1)
sage: b = GF(5)(1)
sage: type(a) is type(b)Truesage: parent(a)Finite Field of size 2
sage: parent(b)Finite Field of size 5>>> from sage.all import *>>> a = GF(Integer(2))(Integer(1))>>> b = GF(Integer(5))(Integer(1))>>> type(a) is type(b)True>>> parent(a)Finite Field of size 2>>> parent(b)Finite Field of size 5

Hence, from an algebraic point of view, the parent of an element is more important than its type.

2.10.4 Conversion versus Coercion
In some cases it is possible to convert an element of one parent structure into an element of a different parent structure.
Such conversion can either be explicit or implicit (this is called coercion).
The reader may know the notions type conversion and type coercion from, e.g., the C programming language. There are
notions of conversion and coercion in Sage as well. But the notions in Sage are centered on parents, not on types. So,
please don’t confuse type conversion in C with conversion in Sage!
We give here a rather brief account. For a detailed description and for information on the implementation, we refer tothe section on coercion in the reference manual and to the thematic tutorial.
There are two extremal positions concerning the possibility of doing arithmetic with elements of different rings:• Different rings are different worlds, and it makes no sense whatsoever to add or multiply elements of different rings;
even 1 + 1/2 makes no sense, since the first summand is an integer and the second a rational.
Or• If an element r1 of one ring R1 can somehow be interpreted in another ring R2, then all arithmetic operations
involving r1 and any element of R2 are allowed. The multiplicative unit exists in all fields and many rings, andthey should all be equal.

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Sage favours a compromise. If P1 and P2 are parent structures and p1 is an element of P1, then the user may explicitly
ask for an interpretation of p1 in P2. This may not be meaningful in all cases or not be defined for all elements of P1,
and it is up to the user to ensure that it makes sense. We refer to this as conversion:
sage: a = GF(2)(1)
sage: b = GF(5)(1)
sage: GF(5)(a) == b
True
sage: GF(2)(b) == a
True>>> from sage.all import *>>> a = GF(Integer(2))(Integer(1))>>> b = GF(Integer(5))(Integer(1))>>> GF(Integer(5))(a) == b
True>>> GF(Integer(2))(b) == aTrue

However, an implicit (or automatic) conversion will only happen if this can be done thoroughly and consistently. Mathematical rigour is essential at that point.
Such an implicit conversion is called coercion. If coercion is defined, then it must coincide with conversion. Two
conditions must be satisfied for a coercion to be defined:
1. A coercion from P1 to P2 must be given by a structure preserving map (e.g., a ring homomorphism). It does not
suffice that some elements of P1 can be mapped to P2, and the map must respect the algebraic structure of P1.
2. The choice of these coercion maps must be consistent: If P3 is a third parent structure, then the composition of
the chosen coercion from P1 to P2 with the coercion from P2 to P3 must coincide with the chosen coercion from
P1 to P3. In particular, if there is a coercion from P1 to P2 and P2 to P1, the composition must be the identity
map of P1.
So, although it is possible to convert each element of GF(2) into GF(5), there is no coercion, since there is no ringhomomorphism between GF(2) and GF(5).
The second aspect - consistency - is a bit more difficult to explain. We illustrate it with multivariate polynomial rings. In
applications, it certainly makes most sense to have name preserving coercions. So, we have:sage: R1.<x,y> = ZZ[]
sage: R2 = ZZ[ y , x ]sage: R2.has_coerce_map_from(R1)
True
sage: R2(x)xsage: R2(y)y>>> from sage.all import *>>> R1 = ZZ[ x, y ]; (x, y,) = R1._first_ngens(2)>>> R2 = ZZ[ y , x ]>>> R2.has_coerce_map_from(R1)
True>>> R2(x)x>>> R2(y)y2.10. Parents, Conversion and Coercion

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If there is no name preserving ring homomorphism, coercion is not defined. However, conversion may still be possible,
namely by mapping ring generators according to their position in the list of generators:
sage: R3 = ZZ[ z , x ]sage: R3.has_coerce_map_from(R1)
False
sage: R3(x)zsage: R3(y)x>>> from sage.all import *>>> R3 = ZZ[ z , x ]>>> R3.has_coerce_map_from(R1)
False>>> R3(x)z>>> R3(y)xBut such position preserving conversions do not qualify as coercion: By composing a name preserving map fromZZ[ x , y ] to ZZ[ y , x ] with a position preserving map from ZZ[ y , x ] to ZZ[ a , b ], a map
would result that is neither name preserving nor position preserving, in violation to consistency.
If there is a coercion, it will be used to compare elements of different rings or to do arithmetic. This is often convenient,but the user should be aware that extending the ==-relation across the borders of different parents may easily result in
overdoing it. For example, while == is supposed to be an equivalence relation on the elements of one ring, this is notnecessarily the case if different rings are involved. For example, 1 in ZZ and in a finite field are considered equal, since
there is a canonical coercion from the integers to any finite field. However, in general there is no coercion between two
different finite fields. Therefore we havesage: GF(5)(1) == 1
True
sage: 1 == GF(2)(1)
True
sage: GF(5)(1) == GF(2)(1)
False
sage: GF(5)(1) != GF(2)(1)
True>>> from sage.all import *>>> GF(Integer(5))(Integer(1)) == Integer(1)True>>> Integer(1) == GF(Integer(2))(Integer(1))
True>>> GF(Integer(5))(Integer(1)) == GF(Integer(2))(Integer(1))
False>>> GF(Integer(5))(Integer(1)) != GF(Integer(2))(Integer(1))
True

Similarly, we havesage: R3(R1.1) == R3.1
True
sage: R1.1 == R3.1
False
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(continued from previous page)

sage: R1.1 != R3.1
True>>> from sage.all import *>>> R3(R1.gen(1)) == R3.gen(1)True>>> R1.gen(1) == R3.gen(1)False>>> R1.gen(1) != R3.gen(1)True

Another consequence of the consistency condition is that coercions can only go from exact rings (e.g., the rationals QQ)
to inexact rings (e.g., real numbers with a fixed precision RR), but not the other way around. The reason is that thecomposition of the coercion from QQ to RR with a conversion from RR to QQ is supposed to be the identity on QQ. But
this is impossible, since some distinct rational numbers may very well be treated equal in RR, as in the following example:
sage: RR(1/10^200+1/10^100) == RR(1/10^100)
True
sage: 1/10^200+1/10^100 == 1/10^100
False>>> from sage.all import *>>> RR(Integer(1)/Integer(10)**Integer(200)+Integer(1)/Integer(10)**Integer(100)) ==␣
˓→RR(Integer(1)/Integer(10)**Integer(100))
True>>> Integer(1)/Integer(10)**Integer(200)+Integer(1)/Integer(10)**Integer(100) ==␣
˓→Integer(1)/Integer(10)**Integer(100)
False

When comparing elements of two parents P1 and P2, it is possible that there is no coercion between the two rings, butthere is a canonical choice of a parent P3 so that both P1 and P2 coerce into P3. In this case, coercion will take place as
well. A typical use case is the sum of a rational number and a polynomial with integer coefficients, yielding a polynomialwith rational coefficients:
sage: P1.<x> = ZZ[]
sage: p = 2*x+3
sage: q = 1/2
sage: parent(p)Univariate Polynomial Ring in x over Integer Ringsage: parent(p+q)
Univariate Polynomial Ring in x over Rational Field>>> from sage.all import *>>> P1 = ZZ[ x ]; (x,) = P1._first_ngens(1)>>> p = Integer(2)*x+Integer(3)>>> q = Integer(1)/Integer(2)>>> parent(p)Univariate Polynomial Ring in x over Integer Ring>>> parent(p+q)
Univariate Polynomial Ring in x over Rational Field

Note that in principle the result would also make sense in the fraction field of ZZ[ x ]. However, Sage tries to choose acanonical common parent that seems to be most natural (QQ[ x ] in our example). If several potential common parents
seem equally natural, Sage will not pick one of them at random, in order to have a reliable result. The mechanisms whichthat choice is based upon is explained in the thematic tutorial.
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No coercion into a common parent will take place in the following example:
sage: R.<x> = QQ[]
sage: S.<y> = QQ[]
sage: x+y
Traceback (most recent call last):...
TypeError: unsupported operand parent(s) for +: Univariate Polynomial Ring in x over␣˓→Rational Field
and Univariate Polynomial Ring in y over Rational Field>>> from sage.all import *>>> R = QQ[ x ]; (x,) = R._first_ngens(1)>>> S = QQ[ y ]; (y,) = S._first_ngens(1)>>> x+yTraceback (most recent call last):...
TypeError: unsupported operand parent(s) for +: Univariate Polynomial Ring in x over␣˓→Rational Field
and Univariate Polynomial Ring in y over Rational Field

The reason is that Sage would not choose one of the potential candidates QQ[ x ][ y ], QQ[ y ][ x ],QQ[ x , y ] or QQ[ y , x ], because all of these four pairwise different structures seem natural common parents,
and there is no apparent canonical choice.

2.11 Finite Groups, Abelian Groups
Sage has some support for computing with permutation groups, finite classical groups (such as 𝑆𝑈 (𝑛, 𝑞)), finite matrix
groups (with your own generators), and abelian groups (even infinite ones). Much of this is implemented using theinterface to GAP.
For example, to create a permutation group, give a list of generators, as in the following example.
sage: G = PermutationGroup([ (1,2,3)(4,5) , (3,4) ])sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]
sage: G.order()120
sage: G.is_abelian()False
sage: G.derived_series()# random-ish output[Permutation Group with generators [(1,2,3)(4,5), (3,4)],
Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]
sage: G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,˓→5)])
sage: G.random_element()# random output(1,5,3)(2,4)
sage: print(latex(G))\langle (3,4), (1,2,3)(4,5) \rangle>>> from sage.all import *>>> G = PermutationGroup([ (1,2,3)(4,5) , (3,4) ])>>> G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]>>> G.order()120(continues on next page)

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(continued from previous page)>>> G.is_abelian()
False>>> G.derived_series()# random-ish output[Permutation Group with generators [(1,2,3)(4,5), (3,4)],
Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]>>> G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,˓→5)])>>> G.random_element()# random output(1,5,3)(2,4)>>> print(latex(G))\langle (3,4), (1,2,3)(4,5) \rangle

You can also obtain the character table (in LaTeX format) in Sage:sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])sage: latex(G.character_table())\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\1 & -\zeta_{3} - 1 & \zeta_{3} & 1 \\1 & \zeta_{3} & -\zeta_{3} - 1 & 1 \\3 & 0 & 0 & -1\end{array}\right)>>> from sage.all import *>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))],␣
˓→[(Integer(1),Integer(2),Integer(3))]])>>> latex(G.character_table())\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\1 & -\zeta_{3} - 1 & \zeta_{3} & 1 \\1 & \zeta_{3} & -\zeta_{3} - 1 & 1 \\3 & 0 & 0 & -1\end{array}\right)

Sage also includes classical and matrix groups over finite fields:
sage: MS = MatrixSpace(GF(7), 2)
sage: gens = [MS([[1,0],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)sage: G.conjugacy_classes_representatives()([1 0] [0 6] [0 4] [6 0] [0 6] [0 4] [0 6] [0 6] [0 6] [4 0][0 1], [1 5], [5 5], [0 6], [1 2], [5 2], [1 0], [1 4], [1 3], [0 2],
[5 0]
[0 3]
)
sage: G = Sp(4,GF(7))
sage: G
Symplectic Group of degree 4 over Finite Field of size 7
sage: G.random_element()# random output[5 5 5 1][0 2 6 3][5 0 1 0][4 6 3 4](continues on next page)

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sage: G.order()276595200>>> from sage.all import *>>> MS = MatrixSpace(GF(Integer(7)), Integer(2))>>> gens = [MS([[Integer(1),Integer(0)],[-Integer(1),Integer(1)]]),MS([[Integer(1),˓→Integer(1)],[Integer(0),Integer(1)]])]>>> G = MatrixGroup(gens)>>> G.conjugacy_classes_representatives()([1 0] [0 6] [0 4] [6 0] [0 6] [0 4] [0 6] [0 6] [0 6] [4 0][0 1], [1 5], [5 5], [0 6], [1 2], [5 2], [1 0], [1 4], [1 3], [0 2],<BLANKLINE>
[5 0]
[0 3]
)
>>> G = Sp(Integer(4),GF(Integer(7)))>>> G
Symplectic Group of degree 4 over Finite Field of size 7>>> G.random_element()# random output[5 5 5 1][0 2 6 3][5 0 1 0][4 6 3 4]>>> G.order()276595200

You can also compute using abelian groups (infinite and finite):sage: F = AbelianGroup(5, [5,5,7,8,9], names= abcde )sage: (a, b, c, d, e) = F.gens()
sage: d * b**2 * c**3b^2*c^3*d
sage: F = AbelianGroup(3,[2]*3); F
Multiplicative Abelian group isomorphic to C2 x C2 x C2
sage: H = AbelianGroup([2,3], names="xy"); H
Multiplicative Abelian group isomorphic to C2 x C3
sage: AbelianGroup(5)Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
sage: AbelianGroup(5).order()+Infinity>>> from sage.all import *>>> F = AbelianGroup(Integer(5), [Integer(5),Integer(5),Integer(7),Integer(8),˓→Integer(9)], names= abcde )>>> (a, b, c, d, e) = F.gens()>>> d * b**Integer(2) * c**Integer(3)b^2*c^3*d>>> F = AbelianGroup(Integer(3),[Integer(2)]*Integer(3)); F
Multiplicative Abelian group isomorphic to C2 x C2 x C2>>> H = AbelianGroup([Integer(2),Integer(3)], names="xy"); H
Multiplicative Abelian group isomorphic to C2 x C3>>> AbelianGroup(Integer(5))
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z>>> AbelianGroup(Integer(5)).order()+Infinity

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2.12 Number Theory
Sage has extensive functionality for number theory. For example, we can do arithmetic in /𝑁

as follows:

sage: R = IntegerModRing(97)sage: a = R(2) / R(3)sage: a33
sage: a.rational_reconstruction()
2/3
sage: b = R(47)sage: b^20052005
50
sage: b.modulus()
97
sage: b.is_square()True>>> from sage.all import *>>> R = IntegerModRing(Integer(97))>>> a = R(Integer(2)) / R(Integer(3))>>> a
33>>> a.rational_reconstruction()
2/3>>> b = R(Integer(47))>>> b**Integer(20052005)
50>>> b.modulus()
97>>> b.is_square()
True

Sage contains standard number theoretic functions. For example,
sage: gcd(515,2005)
5
sage: factor(2005)
5 * 401
sage: c = factorial(25); c
15511210043330985984000000
sage: [valuation(c,p) for p in prime_range(2,23)][22, 10, 6, 3, 2, 1, 1, 1]
sage: next_prime(2005)2011
sage: previous_prime(2005)
2003
sage: divisors(28); sum(divisors(28)); 2*28
[1, 2, 4, 7, 14, 28]
56
56>>> from sage.all import *>>> gcd(Integer(515),Integer(2005))
5>>> factor(Integer(2005))(continues on next page)2.12. Number Theory67

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(continued from previous page)

5 * 401>>> c = factorial(Integer(25)); c
15511210043330985984000000>>> [valuation(c,p) for p in prime_range(Integer(2),Integer(23))][22, 10, 6, 3, 2, 1, 1, 1]>>> next_prime(Integer(2005))
2011>>> previous_prime(Integer(2005))
2003>>> divisors(Integer(28)); sum(divisors(Integer(28))); Integer(2)*Integer(28)[1, 2, 4, 7, 14, 28]
56
56

Perfect!
Sage’s sigma(n,k) function adds up the 𝑘 𝑡ℎ powers of the divisors of n:sage: sigma(28,0); sigma(28,1); sigma(28,2)
6
56
1050
>>> from sage.all import *>>> sigma(Integer(28),Integer(0)); sigma(Integer(28),Integer(1)); sigma(Integer(28),
˓→Integer(2))
6
56
1050

We next illustrate the extended Euclidean algorithm, Euler’s 𝜑-function, and the Chinese remainder theorem:
sage: d,u,v = xgcd(12,15)
sage: d == u*12 + v*15
True
sage: n = 2005
sage: inverse_mod(3,n)
1337
sage: 3 * 1337
4011
sage: prime_divisors(n)[5, 401]
sage: phi = n*prod([1 - 1/p for p in prime_divisors(n)]); phi
1600
sage: euler_phi(n)1600
sage: prime_to_m_part(n, 5)
401>>> from sage.all import *>>> d,u,v = xgcd(Integer(12),Integer(15))>>> d == u*Integer(12) + v*Integer(15)True>>> n = Integer(2005)>>> inverse_mod(Integer(3),n)1337(continues on next page)

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(continued from previous page)>>> Integer(3) * Integer(1337)
4011>>> prime_divisors(n)[5, 401]>>> phi = n*prod([Integer(1) - Integer(1)/p for p in prime_divisors(n)]); phi
1600>>> euler_phi(n)1600>>> prime_to_m_part(n, Integer(5))
401

We next verify something about the 3𝑛 + 1 problem.
sage: n = 2005
sage: for i in range(1000):....:n = 3*odd_part(n) + 1....:
if odd_part(n)==1:....:
print(i)
....:
break
38
>>> from sage.all import *>>> n = Integer(2005)>>> for i in range(Integer(1000)):...
n = Integer(3)*odd_part(n) + Integer(1)...
if odd_part(n)==Integer(1):...
print(i)
...
break
38

Finally we illustrate the Chinese remainder theorem.
sage: x = crt(2, 1, 3, 5); x
11sage: x % 3 # x mod 3 = 2
2
sage: x % 5 # x mod 5 = 1
1
sage: [binomial(13,m) for m in range(14)][1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
sage: [binomial(13,m)%2 for m in range(14)][1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1]
sage: [kronecker(m,13) for m in range(1,13)][1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1]
sage: n = 10000; sum([moebius(m) for m in range(1,n)])-23
sage: Partitions(4).list()[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]>>> from sage.all import *>>> x = crt(Integer(2), Integer(1), Integer(3), Integer(5)); x
11>>> x % Integer(3) # x mod 3 = 2
2>>> x % Integer(5) # x mod 5 = 1
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(continued from previous page)>>> [binomial(Integer(13),m) for m in range(Integer(14))][1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]>>> [binomial(Integer(13),m)%Integer(2) for m in range(Integer(14))][1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1]>>> [kronecker(m,Integer(13)) for m in range(Integer(1),Integer(13))][1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1]>>> n = Integer(10000); sum([moebius(m) for m in range(Integer(1),n)])-23>>> Partitions(Integer(4)).list()[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

2.12.1 𝑝-adic Numbers
The field of 𝑝-adic numbers is implemented in Sage. Note that once a 𝑝-adic field is created, you cannot change its
precision.
sage: K = Qp(11); K
11-adic Field with capped relative precision 20
sage: a = K(211/17); a
4 + 4*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9+ 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16+ 3*11^17 + 11^18 + 7*11^19 + O(11^20)
sage: b = K(3211/11^2); b
10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18)>>> from sage.all import *>>> K = Qp(Integer(11)); K
11-adic Field with capped relative precision 20>>> a = K(Integer(211)/Integer(17)); a4 + 4*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9+ 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16+ 3*11^17 + 11^18 + 7*11^19 + O(11^20)>>> b = K(Integer(3211)/Integer(11)**Integer(2)); b
10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18)

Much work has been done implementing rings of integers in 𝑝-adic fields and number fields. The interested reader isinvited to read Introduction to the p-adics and ask the experts on the sage-support Google group for further details.
A number of related methods are already implemented in the NumberField class.
sage: R.<x> = PolynomialRing(QQ)
sage: K = NumberField(x^3 + x^2 - 2*x + 8,
sage: K.integral_basis()[1, 1/2*a^2 + 1/2*a, a^2]

a )>>> from sage.all import *>>> R = PolynomialRing(QQ, names=( x ,)); (x,) = R._first_ngens(1)>>> K = NumberField(x**Integer(3) + x**Integer(2) - Integer(2)*x + Integer(8),>>> K.integral_basis()[1, 1/2*a^2 + 1/2*a, a^2]a )sage: K.galois_group()
Galois group 3T2 (S3) with order 6 of x^3 + x^2 - 2*x + 8

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Tutorial, Release 10.4>>> from sage.all import *>>> K.galois_group()
Galois group 3T2 (S3) with order 6 of x^3 + x^2 - 2*x + 8
sage: K.polynomial_quotient_ring()
Univariate Quotient Polynomial Ring in a over Rational Field with modulusx^3 + x^2 - 2*x + 8
sage: K.units()(-3*a^2 - 13*a - 13,)sage: K.discriminant()-503
sage: K.class_group()Class group of order 1 of Number Field in a withdefining polynomial x^3 + x^2 - 2*x + 8
sage: K.class_number()1>>> from sage.all import *>>> K.polynomial_quotient_ring()
Univariate Quotient Polynomial Ring in a over Rational Field with modulusx^3 + x^2 - 2*x + 8>>> K.units()(-3*a^2 - 13*a - 13,)>>> K.discriminant()-503>>> K.class_group()Class group of order 1 of Number Field in a withdefining polynomial x^3 + x^2 - 2*x + 8>>> K.class_number()1

2.13 Some More Advanced Mathematics
2.13.1 Algebraic Geometry
You can define arbitrary algebraic varieties in Sage, but sometimes nontrivial functionality is limited to rings over or
finite fields. For example, we compute the union of two affine plane curves, then recover the curves as the irreducible
components of the union.
sage: x, y = AffineSpace(2, QQ, xy ).gens()
sage: C2 = Curve(x^2 + y^2 - 1)sage: C3 = Curve(x^3 + y^3 - 1)sage: D = C2 + C3
sage: D
Affine Plane Curve over Rational Field defined byx^5 + x^3*y^2 + x^2*y^3 + y^5 - x^3 - y^3 - x^2 - y^2 + 1
sage: D.irreducible_components()[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:x^2 + y^2 - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:x^3 + y^3 - 1]

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Tutorial, Release 10.4>>> from sage.all import *>>> x, y = AffineSpace(Integer(2), QQ, xy ).gens()>>> C2 = Curve(x**Integer(2) + y**Integer(2) - Integer(1))>>> C3 = Curve(x**Integer(3) + y**Integer(3) - Integer(1))>>> D = C2 + C3>>> D
Affine Plane Curve over Rational Field defined byx^5 + x^3*y^2 + x^2*y^3 + y^5 - x^3 - y^3 - x^2 - y^2 + 1>>> D.irreducible_components()[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:x^2 + y^2 - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:x^3 + y^3 - 1]

We can also find all points of intersection of the two curves by intersecting them and computing the irreducible components.
sage: V = C2.intersection(C3)sage: V.irreducible_components()[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:y,x - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:y - 1,x,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x + y + 2,
2*y^2 + 4*y + 3]>>> from sage.all import *>>> V = C2.intersection(C3)>>> V.irreducible_components()[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:y,x - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:y - 1,x,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x + y + 2,
2*y^2 + 4*y + 3]

Thus, e.g., (1, 0) and (0, 1) are on both curves (visibly clear), as are certain (quadratic) points whose 𝑦 coordinates satisfy
2𝑦 2 + 4𝑦 + 3 = 0.
Sage can compute the toric ideal of the twisted cubic in projective 3 space:sage: R.<a,b,c,d> = PolynomialRing(QQ, 4)
sage: I = ideal(b^2-a*c, c^2-b*d, a*d-b*c)
sage: F = I.groebner_fan(); F
Groebner fan of the ideal:(continues on next page)

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Ideal (b^2 - a*c, c^2 - b*d, -b*c + a*d) of Multivariate Polynomial Ringin a, b, c, d over Rational Field
sage: F.reduced_groebner_bases ()[[-c^2 + b*d, -b*c + a*d, -b^2 + a*c],[-b*c + a*d, -c^2 + b*d, b^2 - a*c],[-c^3 + a*d^2, -c^2 + b*d, b*c - a*d, b^2 - a*c],[-c^2 + b*d, b^2 - a*c, b*c - a*d, c^3 - a*d^2],[-b*c + a*d, -b^2 + a*c, c^2 - b*d],[-b^3 + a^2*d, -b^2 + a*c, c^2 - b*d, b*c - a*d],[-b^2 + a*c, c^2 - b*d, b*c - a*d, b^3 - a^2*d],[c^2 - b*d, b*c - a*d, b^2 - a*c]]
sage: F.polyhedralfan()
Polyhedral fan in 4 dimensions of dimension 4>>> from sage.all import *>>> R = PolynomialRing(QQ, Integer(4), names=( a , b , c , d ,)); (a, b, c, d,) =␣
˓→R._first_ngens(4)>>> I = ideal(b**Integer(2)-a*c, c**Integer(2)-b*d, a*d-b*c)>>> F = I.groebner_fan(); F
Groebner fan of the ideal:
Ideal (b^2 - a*c, c^2 - b*d, -b*c + a*d) of Multivariate Polynomial Ringin a, b, c, d over Rational Field>>> F.reduced_groebner_bases ()[[-c^2 + b*d, -b*c + a*d, -b^2 + a*c],[-b*c + a*d, -c^2 + b*d, b^2 - a*c],[-c^3 + a*d^2, -c^2 + b*d, b*c - a*d, b^2 - a*c],[-c^2 + b*d, b^2 - a*c, b*c - a*d, c^3 - a*d^2],[-b*c + a*d, -b^2 + a*c, c^2 - b*d],[-b^3 + a^2*d, -b^2 + a*c, c^2 - b*d, b*c - a*d],[-b^2 + a*c, c^2 - b*d, b*c - a*d, b^3 - a^2*d],[c^2 - b*d, b*c - a*d, b^2 - a*c]]>>> F.polyhedralfan()
Polyhedral fan in 4 dimensions of dimension 4

2.13.2 Elliptic Curves
Elliptic curve functionality includes most of the elliptic curve functionality of PARI, access to the data in Cremona’s
online tables (this requires an optional database package), the functionality of mwrank, i.e., 2-descents with computationof the full Mordell-Weil group, the SEA algorithm, computation of all isogenies, much new code for curves over , andsome of Denis Simon’s algebraic descent software.
The command EllipticCurve for creating an elliptic curve has many forms:• EllipticCurve([𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 , 𝑎6 ]): Returns the elliptic curve𝑦 2 + 𝑎1 𝑥𝑦 + 𝑎3 𝑦 = 𝑥3 + 𝑎2 𝑥2 + 𝑎4 𝑥 + 𝑎6 ,where the 𝑎𝑖 ’s are coerced into the parent of 𝑎1 . If all the 𝑎𝑖 have parent , they are coerced into.

• EllipticCurve([𝑎4 , 𝑎6 ]): Same as above, but 𝑎1 = 𝑎2 = 𝑎3 = 0.
• EllipticCurve(label): Returns the elliptic curve over from the Cremona database with the given (new!) Cremonalabel. The label is a string, such as "11a" or "37b2". The letter must be lower case (to distinguish it from theold labeling).
• EllipticCurve(j): Returns an elliptic curve with 𝑗-invariant 𝑗.

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• EllipticCurve(R, [𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 , 𝑎6 ]): Create the elliptic curve over a ring 𝑅 with given 𝑎𝑖 ’s as above.
We illustrate each of these constructors:
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve([1,2])
Elliptic Curve defined by y^2= x^3 + x + 2 over Rational Field

sage: EllipticCurve( 37a )Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: EllipticCurve_from_j(1)Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
sage: EllipticCurve(GF(5), [0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5>>> from sage.all import *>>> EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field>>> EllipticCurve([GF(Integer(5))(Integer(0)),Integer(0),Integer(1),-Integer(1),˓→Integer(0)])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5>>> EllipticCurve([Integer(1),Integer(2)])
Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field>>> EllipticCurve( 37a )Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field>>> EllipticCurve_from_j(Integer(1))
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field>>> EllipticCurve(GF(Integer(5)), [Integer(0),Integer(0),Integer(1),-Integer(1),˓→Integer(0)])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5

The pair (0, 0) is a point on the elliptic curve 𝐸 defined by 𝑦 2 +𝑦 = 𝑥3 −𝑥. To create this point in Sage type E([0,0]).
Sage can add points on such an elliptic curve (recall elliptic curves support an additive group structure where the point at
infinity is the zero element and three co-linear points on the curve add to zero):sage: E = EllipticCurve([0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E([0,0])
sage: P + P(1 : 0 : 1)sage: 10*P
(161/16 : -2065/64 : 1)sage: 20*P
(683916417/264517696 : -18784454671297/4302115807744 : 1)sage: E.conductor()(continues on next page)

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(continued from previous page)37>>> from sage.all import *>>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])>>> E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field>>> P = E([Integer(0),Integer(0)])>>> P + P
(1 : 0 : 1)>>> Integer(10)*P(161/16 : -2065/64 : 1)>>> Integer(20)*P(683916417/264517696 : -18784454671297/4302115807744 : 1)>>> E.conductor()
37

The elliptic curves over the complex numbers are parameterized by the 𝑗-invariant. Sage computes 𝑗-invariant as follows:
sage: E = EllipticCurve([0,0,0,-4,2]); E
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
sage: E.conductor()2368
sage: E.j_invariant()
110592/37>>> from sage.all import *>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),-Integer(4),Integer(2)]); E
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field>>> E.conductor()
2368>>> E.j_invariant()
110592/37

If we make a curve with the same 𝑗-invariant as that of 𝐸, it need not be isomorphic to 𝐸. In the following example, thecurves are not isomorphic because their conductors are different.
sage: F = EllipticCurve_from_j(110592/37)
sage: F.conductor()
37>>> from sage.all import *>>> F = EllipticCurve_from_j(Integer(110592)/Integer(37))>>> F.conductor()
37

However, the twist of 𝐹 by 2 gives an isomorphic curve.
sage: G = F.quadratic_twist(2); G
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
sage: G.conductor()2368
sage: G.j_invariant()
110592/37

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Tutorial, Release 10.4>>> from sage.all import *>>> G = F.quadratic_twist(Integer(2)); G
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field>>> G.conductor()
2368>>> G.j_invariant()
110592/37

We can compute the coefficients 𝑎𝑛 of the 𝐿-series or modular form
computation uses the PARI C-library:∑︀∞

𝑛=0 𝑎𝑛 𝑞𝑛attached to the elliptic curve. This

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.anlist(30)[0, 1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4,
3, 10, 2, 0, -1, 4, -9, -2, 6, -12]
sage: v = E.anlist(10000)>>> from sage.all import *>>> E = EllipticCurve([Integer(0),Integer(0),Integer(1),-Integer(1),Integer(0)])>>> E.anlist(Integer(30))[0, 1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4,
3, 10, 2, 0, -1, 4, -9, -2, 6, -12]>>> v = E.anlist(Integer(10000))

It only takes a second to compute all 𝑎𝑛 for 𝑛 ≤ 105 :sage: %time v = E.anlist(100000)
CPU times: user 0.98 s, sys: 0.06 s, total: 1.04 s
Wall time: 1.06>>> from sage.all import *>>> %time v = E.anlist(Integer(100000))
CPU times: user 0.98 s, sys: 0.06 s, total: 1.04 s
Wall time: 1.06

Elliptic curves can be constructed using their Cremona labels. This pre-loads the elliptic curve with information about its
rank, Tamagawa numbers, regulator, etc.
sage: E = EllipticCurve("37b2")
sage: E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational
Field
sage: E = EllipticCurve("389a")sage: E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: E.rank()
2sage: E = EllipticCurve("5077a")sage: E.rank()
3>>> from sage.all import *>>> E = EllipticCurve("37b2")>>> E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational
Field(continues on next page)

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(continued from previous page)>>> E = EllipticCurve("389a")>>> E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x>>> E.rank()
2>>> E = EllipticCurve("5077a")>>> E.rank()
3over Rational Field

We can also access the Cremona database directly.
sage: db = sage.databases.cremona.CremonaDatabase()sage: db.curves(37){ a1 : [[0, 0, 1, -1, 0], 1, 1], b1 : [[0, 1, 1, -23, -50], 0, 3]}
sage: db.allcurves(37){ a1 : [[0, 0, 1, -1, 0], 1, 1],b1 : [[0, 1, 1, -23, -50], 0, 3],b2 : [[0, 1, 1, -1873, -31833], 0, 1],b3 : [[0, 1, 1, -3, 1], 0, 3]}>>> from sage.all import *>>> db = sage.databases.cremona.CremonaDatabase()>>> db.curves(Integer(37)){ a1 : [[0, 0, 1, -1, 0], 1, 1], b1 : [[0, 1, 1, -23, -50], 0, 3]}>>> db.allcurves(Integer(37)){ a1 : [[0, 0, 1, -1, 0], 1, 1],b1 : [[0, 1, 1, -23, -50], 0, 3],b2 : [[0, 1, 1, -1873, -31833], 0, 1],b3 : [[0, 1, 1, -3, 1], 0, 3]}

The objects returned from the database are not of type EllipticCurve. They are elements of a database and have acouple of fields, and that’s it. There is a small version of Cremona’s database, which is distributed by default with Sage,
and contains limited information about elliptic curves of conductor ≤ 10000. There is also a large optional version, whichcontains extensive data about all curves of conductor up to 120000 (as of October 2005). There is also a huge (2GB)
optional database package for Sage that contains the hundreds of millions of elliptic curves in the Stein-Watkins database.

2.13.3 Dirichlet Characters
A Dirichlet character is the extension of a homomorphism ( /𝑁 )* → 𝑅* , for some ring 𝑅, to the map
by sending those integers 𝑥 with gcd(𝑁, 𝑥) > 1 to 0.

→ 𝑅 obtained

sage: G = DirichletGroup(12)sage: G.list()[Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1,
Dirichlet character modulo 12 of conductor 12 mapping 7 |--> -1, 5 |--> -1]
sage: G.gens()(Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1)
sage: len(G)4

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Tutorial, Release 10.4>>> from sage.all import *>>> G = DirichletGroup(Integer(12))>>> G.list()[Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1,
Dirichlet character modulo 12 of conductor 12 mapping 7 |--> -1, 5 |--> -1]>>> G.gens()(Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1)>>> len(G)4Having created the group, we next create an element and compute with it.
sage: G = DirichletGroup(21)sage: chi = G.1; chi
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6
sage: chi.values()[0, 1, zeta6 - 1, 0, -zeta6, -zeta6 + 1, 0, 0, 1, 0, zeta6, -zeta6, 0, -1,
0, 0, zeta6 - 1, zeta6, 0, -zeta6 + 1, -1]
sage: chi.conductor()7
sage: chi.modulus()21
sage: chi.order()6sage: chi(19)-zeta6 + 1
sage: chi(40)-zeta6 + 1>>> from sage.all import *>>> G = DirichletGroup(Integer(21))>>> chi = G.gen(1); chi
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6>>> chi.values()[0, 1, zeta6 - 1, 0, -zeta6, -zeta6 + 1, 0, 0, 1, 0, zeta6, -zeta6, 0, -1,
0, 0, zeta6 - 1, zeta6, 0, -zeta6 + 1, -1]>>> chi.conductor()
7>>> chi.modulus()21>>> chi.order()6>>> chi(Integer(19))-zeta6 + 1>>> chi(Integer(40))-zeta6 + 1

It is also possible to compute the action of the Galois group Gal( (𝜁𝑁 )/ ) on these characters, as well as the direct
product decomposition corresponding to the factorization of the modulus.
sage: chi.galois_orbit()[Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> -zeta6 + 1,
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6](continues on next page)

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sage: go = G.galois_orbits()
sage: [len(orbit) for orbit in go][1, 2, 2, 1, 1, 2, 2, 1]
sage: G.decomposition()[Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 6 and␣
˓→degree 2,Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and␣
˓→degree 2]>>> from sage.all import *>>> chi.galois_orbit()[Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> -zeta6 + 1,
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6]>>> go = G.galois_orbits()>>> [len(orbit) for orbit in go][1, 2, 2, 1, 1, 2, 2, 1]>>> G.decomposition()[Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 6 and␣
˓→degree 2,Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and␣
˓→degree 2]

Next, we construct the group of Dirichlet characters mod 20, but with values in(𝑖):sage: K.<i> = NumberField(x^2+1)
sage: G = DirichletGroup(20,K)
sage: GGroup of Dirichlet characters modulo 20 with values in Number Field in i with␣
˓→defining polynomial x^2 + 1>>> from sage.all import *>>> K = NumberField(x**Integer(2)+Integer(1), names=( i ,)); (i,) = K._first_ngens(1)>>> G = DirichletGroup(Integer(20),K)>>> G
Group of Dirichlet characters modulo 20 with values in Number Field in i with␣
˓→defining polynomial x^2 + 1

We next compute several invariants of G:
sage: G.gens()(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> i)sage: G.unit_gens()(11, 17)
sage: G.zeta()isage: G.zeta_order()4

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Tutorial, Release 10.4>>> from sage.all import *>>> G.gens()(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> i)>>> G.unit_gens()(11, 17)>>> G.zeta()i>>> G.zeta_order()4

In this example we create a Dirichlet character with values in a number field. We explicitly specify the choice of root ofunity by the third argument to DirichletGroup below.
sage: x = polygen(QQ, x )sage: K = NumberField(x^4 + 1, a ); a = K.0
sage: b = K.gen(); a == b
True
sage: K
Number Field in a with defining polynomial x^4 + 1
sage: G = DirichletGroup(5, K, a); GGroup of Dirichlet characters modulo 5 with values in the group of order 8 generated␣
˓→by a in Number Field in a with defining polynomial x^4 + 1
sage: chi = G.0; chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2
sage: [(chi^i)(2) for i in range(4)][1, a^2, -1, -a^2]>>> from sage.all import *>>> x = polygen(QQ, x )>>> K = NumberField(x**Integer(4) + Integer(1), a ); a = K.gen(0)>>> b = K.gen(); a == b
True
>>> K
Number Field in a with defining polynomial x^4 + 1>>> G = DirichletGroup(Integer(5), K, a); GGroup of Dirichlet characters modulo 5 with values in the group of order 8 generated␣
˓→by a in Number Field in a with defining polynomial x^4 + 1>>> chi = G.gen(0); chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2>>> [(chi**i)(Integer(2)) for i in range(Integer(4))][1, a^2, -1, -a^2]Here NumberField(x^4 + 1, a ) tells Sage to use the symbol “a” in printing what K is (a Number Field ina with defining polynomial 𝑥4 + 1). The name “a” is undeclared at this point. Once a = K.0 (or equivalently a =K.gen()) is evaluated, the symbol “a” represents a root of the generating polynomial 𝑥4 + 1.

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2.13.4 Modular Forms
Sage can do some computations related to modular forms, including dimensions, computing spaces of modular symbols,
Hecke operators, and decompositions.
There are several functions available for computing dimensions of spaces of modular forms. For example,
sage: from sage.modular.dims import dimension_cusp_forms
sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma0(1),12)1
sage: dimension_cusp_forms(Gamma1(389),2)
6112>>> from sage.all import *>>> from sage.modular.dims import dimension_cusp_forms>>> dimension_cusp_forms(Gamma0(Integer(11)),Integer(2))
1>>> dimension_cusp_forms(Gamma0(Integer(1)),Integer(12))
1>>> dimension_cusp_forms(Gamma1(Integer(389)),Integer(2))
6112

Next we illustrate computation of Hecke operators on a space of modular symbols of level 1 and weight 12.
sage: M = ModularSymbols(1,12)
sage: M.basis()([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
sage: t2 = M.T(2)sage: t2
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)of weight 12 with sign 0 over Rational Field
sage: t2.matrix()
[ -24
0
0]
[
0 -24
0]
[4860
0 2049]sage: f = t2.charpoly( x ); fx^3 - 2001*x^2 - 97776*x - 1180224
sage: factor(f)(x - 2049) * (x + 24)^2
sage: M.T(11).charpoly( x ).factor()(x - 285311670612) * (x - 534612)^2>>> from sage.all import *>>> M = ModularSymbols(Integer(1),Integer(12))>>> M.basis()([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])>>> t2 = M.T(Integer(2))>>> t2
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)of weight 12 with sign 0 over Rational Field>>> t2.matrix()
[ -24
0
0]
[
0 -24
0]
[4860
0 2049]>>> f = t2.charpoly( x ); f
(continues on next page)

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(continued from previous page)x^3 - 2001*x^2 - 97776*x - 1180224>>> factor(f)(x - 2049) * (x + 24)^2>>> M.T(Integer(11)).charpoly( x ).factor()(x - 285311670612) * (x - 534612)^2

We can also create spaces for Γ0 (𝑁 ) and Γ1 (𝑁 ).
sage: ModularSymbols(11,2)
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: ModularSymbols(Gamma1(11),2)
Modular Symbols space of dimension 11 for Gamma_1(11) of weight 2 withsign 0 over Rational Field>>> from sage.all import *>>> ModularSymbols(Integer(11),Integer(2))
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field>>> ModularSymbols(Gamma1(Integer(11)),Integer(2))
Modular Symbols space of dimension 11 for Gamma_1(11) of weight 2 withsign 0 over Rational Field

Let’s compute some characteristic polynomials and 𝑞-expansions.
sage: M = ModularSymbols(Gamma1(11),2)
sage: M.T(2).charpoly( x )x^11 - 8*x^10 + 20*x^9 + 10*x^8 - 145*x^7 + 229*x^6 + 58*x^5 - 360*x^4+ 70*x^3 - 515*x^2 + 1804*x - 1452
sage: M.T(2).charpoly( x ).factor()(x - 3) * (x + 2)^2 * (x^4 - 7*x^3 + 19*x^2 - 23*x + 11)* (x^4 - 2*x^3 + 4*x^2 + 2*x + 11)
sage: S = M.cuspidal_submodule()
sage: S.T(2).matrix()[-2 0][ 0 -2]sage: S.q_expansion_basis(10)[q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10)]>>> from sage.all import *>>> M = ModularSymbols(Gamma1(Integer(11)),Integer(2))>>> M.T(Integer(2)).charpoly( x )x^11 - 8*x^10 + 20*x^9 + 10*x^8 - 145*x^7 + 229*x^6 + 58*x^5 - 360*x^4+ 70*x^3 - 515*x^2 + 1804*x - 1452>>> M.T(Integer(2)).charpoly( x ).factor()(x - 3) * (x + 2)^2 * (x^4 - 7*x^3 + 19*x^2 - 23*x + 11)* (x^4 - 2*x^3 + 4*x^2 + 2*x + 11)>>> S = M.cuspidal_submodule()>>> S.T(Integer(2)).matrix()[-2 0][ 0 -2]>>> S.q_expansion_basis(Integer(10))[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10)(continues on next page)

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We can even compute spaces of modular symbols with character.
sage: G = DirichletGroup(13)sage: e = G.0^2
sage: M = ModularSymbols(e,2); M
Modular Symbols space of dimension 4 and level 13, weight 2, character[zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: M.T(2).charpoly( x ).factor()(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)^2
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular Symbols space ofdimension 4 and level 13, weight 2, character [zeta6], sign 0, over
Cyclotomic Field of order 6 and degree 2
sage: S.T(2).charpoly( x ).factor()(x + zeta6 + 1)^2
sage: S.q_expansion_basis(10)[q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5+ (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)]>>> from sage.all import *>>> G = DirichletGroup(Integer(13))>>> e = G.gen(0)**Integer(2)>>> M = ModularSymbols(e,Integer(2)); M
Modular Symbols space of dimension 4 and level 13, weight 2, character[zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2>>> M.T(Integer(2)).charpoly( x ).factor()(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)^2>>> S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular Symbols space ofdimension 4 and level 13, weight 2, character [zeta6], sign 0, over
Cyclotomic Field of order 6 and degree 2>>> S.T(Integer(2)).charpoly( x ).factor()(x + zeta6 + 1)^2>>> S.q_expansion_basis(Integer(10))[
q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5+ (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)]

Here is another example of how Sage can compute the action of Hecke operators on a space of modular forms.
sage: T = ModularForms(Gamma0(11),2)
sage: T
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) ofweight 2 over Rational Field
sage: T.degree()2
sage: T.level()11
sage: T.group()
Congruence Subgroup Gamma0(11)sage: T.dimension()(continues on next page)

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(continued from previous page)2
sage: T.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 forCongruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: T.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: M = ModularSymbols(11); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: M.weight()
2
sage: M.basis()((1,0), (1,8), (1,9))
sage: M.sign()
0>>> from sage.all import *>>> T = ModularForms(Gamma0(Integer(11)),Integer(2))>>> T
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) ofweight 2 over Rational Field>>> T.degree()2>>> T.level()
11>>> T.group()
Congruence Subgroup Gamma0(11)>>> T.dimension()2>>> T.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 forCongruence Subgroup Gamma0(11) of weight 2 over Rational Field>>> T.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field>>> M = ModularSymbols(Integer(11)); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field>>> M.weight()
2>>> M.basis()((1,0), (1,8), (1,9))>>> M.sign()
0

Let 𝑇𝑝 denote the usual Hecke operators (𝑝 prime). How do the Hecke operators 𝑇2 , 𝑇3 , 𝑇5 act on the space of modular
symbols?
sage: M.T(2).matrix()[ 3 0 -1]
[ 0 -2 0][ 0 0 -2]
sage: M.T(3).matrix()[ 4 0 -1]
[ 0 -1 0](continues on next page)

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(continued from previous page)[ 0 0 -1]
sage: M.T(5).matrix()[ 6 0 -1]
[ 0 1 0][ 0 0 1]>>> from sage.all import *>>> M.T(Integer(2)).matrix()[ 3 0 -1]
[ 0 -2 0][ 0 0 -2]>>> M.T(Integer(3)).matrix()[ 4 0 -1]
[ 0 -1 0][ 0 0 -1]>>> M.T(Integer(5)).matrix()[ 6 0 -1]
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CHAPTER

THREE

THE INTERACTIVE SHELL

In most of this tutorial, we assume you start the Sage interpreter using the sage command. This starts a customized
version of the IPython shell, and imports many functions and classes, so they are ready to use from the command prompt.
Further customization is possible by editing the $SAGE_ROOT/ipythonrc file. Upon starting Sage, you get output
similar to the following:┌────────────────────────────────────────────────────────────────────┐│ SageMath version 9.7, Release Date: 2022-01-10││ Using Python 3.10.4. Type "help()" for help.
│└────────────────────────────────────────────────────────────────────┘

sage:

To quit Sage either press Ctrl-D or type quit or exit.
sage: quit
Exiting Sage (CPU time 0m0.00s, Wall time 0m0.89s)>>> from sage.all import *>>> quit
Exiting Sage (CPU time 0m0.00s, Wall time 0m0.89s)

The wall time is the time that elapsed on the clock hanging from your wall. This is relevant, since CPU time does nottrack time used by subprocesses like GAP or Singular.
(Avoid killing a Sage process with kill -9 from a terminal, since Sage might not kill child processes, e.g., Maple
processes, or cleanup temporary files from $HOME/.sage/tmp.)

3.1 Your Sage Session
The session is the sequence of input and output from when you start Sage until you quit. Sage logs all Sage input,
via IPython. In fact, if you’re using the interactive shell (not the notebook interface), then at any point you may type
%history (or %hist) to get a listing of all input lines typed so far. You can type ? at the Sage prompt to find out
more about IPython, e.g., “IPython offers numbered prompts … with input and output caching. All input is saved andcan be retrieved as variables (besides the usual arrow key recall). The following GLOBAL variables always exist (so don’t
overwrite them!)”:_: previous input (interactive shell and notebook)__: next previous input (interactive shell only)_oh : list of all inputs (interactive shell only)87Tutorial, Release 10.4

Here is an example:
sage: factor(100)_1 = 2^2 * 5^2sage: kronecker_symbol(3,5)_2 = -1
sage: %hist#This only works from the interactive shell, not the notebook.
1: factor(100)2: kronecker_symbol(3,5)
3: %hist
sage: _oh
_4 = {1: 2^2 * 5^2, 2: -1}
sage: _i1
_5 = factor(ZZ(100))\n
sage: eval(_i1)_6 = 2^2 * 5^2sage: %hist
1: factor(100)2: kronecker_symbol(3,5)
3: %hist
4: _oh5: _i1
6: eval(_i1)
7: %hist>>> from sage.all import *>>> factor(Integer(100))_1 = 2^2 * 5^2>>> kronecker_symbol(Integer(3),Integer(5))_2 = -1>>> %hist#This only works from the interactive shell, not the notebook.
1: factor(100)2: kronecker_symbol(3,5)
3: %hist>>> _oh_4 = {1: 2^2 * 5^2, 2: -1}>>> _i1_5 = factor(ZZ(100))\n>>> eval(_i1)_6 = 2^2 * 5^2>>> %hist
1: factor(100)2: kronecker_symbol(3,5)
3: %hist
4: _oh5: _i1
6: eval(_i1)
7: %hist

We omit the output numbering in the rest of this tutorial and the other Sage documentation.
You can also store a list of input from session in a macro for that session.
sage: E = EllipticCurve([1,2,3,4,5])
sage: M = ModularSymbols(37)sage: %hist
1: E = EllipticCurve([1,2,3,4,5])
2: M = ModularSymbols(37)(continues on next page)

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(continued from previous page)

3: %hist
sage: %macro em 1-2
Macro em created. To execute, type its name (without quotes).
>>> from sage.all import *>>> E = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])>>> M = ModularSymbols(Integer(37))>>> %hist
1: E = EllipticCurve([1,2,3,4,5])
2: M = ModularSymbols(37)3: %hist>>> %macro em Integer(1)-Integer(2)Macro em created. To execute, type its name (without quotes).
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
Rational Field
sage: E = 5
sage: M = Nonesage: em
Executing Macro...
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
Rational Field>>> from sage.all import *>>> E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
Rational Field>>> E = Integer(5)>>> M = None>>> emExecuting Macro...
>>> E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
Rational Field

When using the interactive shell, any UNIX shell command can be executed from Sage by prefacing it by an exclamation
point !. For example,
sage: !ls
auto example.sage glossary.texttmp

tut.log

tut.tex>>> from sage.all import *>>> !ls
auto example.sage glossary.texttmp

tut.log

tut.texreturns the listing of the current directory.
The PATH has the Sage bin directory at the front, so if you run gp, gap, singular, maxima, etc., you get the versions
included with Sage.
sage: !gp
Reading GPRC: /etc/gprc ...Done.
(continues on next page)

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(continued from previous page)

GP/PARI CALCULATOR Version 2.2.11 (alpha)i686 running linux (ix86/GMP-4.1.4 kernel) 32-bit version...
sage: !singular
SINGULARA Computer Algebra System for Polynomial Computations//
0<
by: G.-M. Greuel, G. Pfister, H. Schoenemann\
FB Mathematik der Universitaet, D-67653 Kaiserslautern\

Development
version 3-0-1
October 2005>>> from sage.all import *>>> !gp
Reading GPRC: /etc/gprc ...Done.
GP/PARI CALCULATOR Version 2.2.11 (alpha)i686 running linux (ix86/GMP-4.1.4 kernel) 32-bit version...
>>> !singular
SINGULARA Computer Algebra System for Polynomial Computations//
0<
by: G.-M. Greuel, G. Pfister, H. Schoenemann\
FB Mathematik der Universitaet, D-67653 Kaiserslautern\

Development
version 3-0-1
October 2005

3.2 Logging Input and Output
Logging your Sage session is not the same as saving it (see Saving and Loading Complete Sessions for that). To log input(and optionally output) use the logstart command. Type logstart? for more details. You can use this command
to log all input you type, all output, and even play back that input in a future session (by simply reloading the log file).
was@form:~$ sage
┌────────────────────────────────────────────────────────────────────┐│ SageMath version 9.7, Release Date: 2022-01-10││ Using Python 3.10.4. Type "help()" for help.
│└────────────────────────────────────────────────────────────────────┘
sage: logstart setup
Activating auto-logging. Current session state plus future input saved.
Filename: setupMode: backup
Output logging : False
Timestamping: FalseState: activesage: E = EllipticCurve([1,2,3,4,5]).minimal_model()sage: F = QQ^3sage: x,y = QQ[ x,y ].gens()
sage: G = E.gens()sage:
Exiting Sage (CPU time 0m0.61s, Wall time 0m50.39s).
was@form:~$ sage
┌────────────────────────────────────────────────────────────────────┐│ SageMath version 9.7, Release Date: 2022-01-10││ Using Python 3.10.4. Type "help()" for help.
│(continues on next page)

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(continued from previous page)└────────────────────────────────────────────────────────────────────┘
sage: load("setup")
Loading log file <setup> one line at a time...
Finished replaying log file <setup>
sage: E
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 + 4*x + 3 over Rational
Field
sage: x*y
x*y
sage: G[(2 : 3 : 1)]

If you use Sage in the Linux KDE terminal konsole then you can save your session as follows: after starting Sagein konsole, select “settings”, then “history…”, then “set unlimited”. When you are ready to save your session, select“edit” then “save history as…” and type in a name to save the text of your session to your computer. After saving thisfile, you could then load it into an editor, such as xemacs, and print it.

3.3 Paste Ignores Prompts
Suppose you are reading a session of Sage or Python computations and want to copy them into Sage. But there areannoying >>> or sage: prompts to worry about. In fact, you can copy and paste an example, including the prompts if
you want, into Sage. In other words, by default the Sage parser strips any leading >>> or sage: prompt before passing
it to Python. For example,
sage: 2^101024
sage: sage: sage: 2^101024
sage: >>> 2^101024>>> from sage.all import *>>> Integer(2)**Integer(10)1024>>> sage: sage: Integer(2)**Integer(10)1024>>> >>> Integer(2)**Integer(10)1024

3.4 Timing Commands
If you place the %time command at the beginning of an input line, the time the command takes to run will be displayed
after the output. For example, we can compare the running time for a certain exponentiation operation in several ways.
The timings below will probably be much different on your computer, or even between different versions of Sage. First,
native Python:sage: %time a = int(1938)^int(99484)
CPU times: user 0.66 s, sys: 0.00 s, total: 0.66 s
Wall time: 0.66

3.3. Paste Ignores Prompts

91Tutorial, Release 10.4>>> from sage.all import *>>> %time a = int(Integer(1938))**int(Integer(99484))
CPU times: user 0.66 s, sys: 0.00 s, total: 0.66 s
Wall time: 0.66

This means that 0.66 seconds total were taken, and the “Wall time”, i.e., the amount of time that elapsed on your wall
clock, is also 0.66 seconds. If your computer is heavily loaded with other programs, the wall time may be much larger
than the CPU time.
It’s also possible to use the timeit function to try to get timing over a large number of iterations of a command. This
gives slightly different information, and requires the input of a string with the command you want to time.
sage: timeit("int(1938)^int(99484)")
5 loops, best of 3: 44.8 ms per loop>>> from sage.all import *>>> timeit("int(1938)^int(99484)")
5 loops, best of 3: 44.8 ms per loop

Next we time exponentiation using the native Sage Integer type, which is implemented (in Cython) using the GMP library:sage: %time a = 1938^99484
CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
Wall time: 0.04>>> from sage.all import *>>> %time a = Integer(1938)**Integer(99484)
CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
Wall time: 0.04

Using the PARI C-library interface:sage: %time a = pari(1938)^pari(99484)
CPU times: user 0.05 s, sys: 0.00 s, total: 0.05 s
Wall time: 0.05>>> from sage.all import *>>> %time a = pari(Integer(1938))**pari(Integer(99484))
CPU times: user 0.05 s, sys: 0.00 s, total: 0.05 s
Wall time: 0.05

GMP is better, but only slightly (as expected, since the version of PARI built for Sage uses GMP for integer arithmetic).
You can also time a block of commands using the cputime command, as illustrated below:
sage: t = cputime()
sage: a = int(1938)^int(99484)
sage: b = 1938^99484
sage: c = pari(1938)^pari(99484)
sage: cputime(t)0.64# somewhat random output>>> from sage.all import *>>> t = cputime()>>> a = int(Integer(1938))**int(Integer(99484))>>> b = Integer(1938)**Integer(99484)(continues on next page)

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(continued from previous page)>>> c = pari(Integer(1938))**pari(Integer(99484))>>> cputime(t)# somewhat random output
0.64
sage: cputime?
...
Return the time in CPU second since Sage started, or with optional
argument t, return the time since time t.
INPUT:
t -- (optional) float, time in CPU seconds
OUTPUT:
float -- time in CPU seconds>>> from sage.all import *>>> cputime?
...
Return the time in CPU second since Sage started, or with optional
argument t, return the time since time t.
INPUT:
t -- (optional) float, time in CPU seconds
OUTPUT:
float -- time in CPU seconds

The walltime command behaves just like the cputime command, except that it measures wall time.
We can also compute the above power in some of the computer algebra systems that Sage includes. In each case we
execute a trivial command in the system, in order to start up the server for that program. The most relevant time is thewall time. However, if there is a significant difference between the wall time and the CPU time then this may indicate aperformance issue worth looking into.
sage: time 1938^99484;
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01
sage: gp(0)0sage: time g = gp( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.04
sage: maxima(0)0sage: time g = maxima( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.30
sage: kash(0)0sage: time g = kash( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.04
sage: mathematica(0)0sage: time g = mathematica( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.03
sage: maple(0)0sage: time g = maple( 1938^99484 )(continues on next page)

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CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.11
sage: gap(0)0sage: time g = gap.eval( 1938^99484;; )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 1.02>>> from sage.all import *>>> time Integer(1938)**Integer(99484);
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.01>>> gp(Integer(0))
0>>> time g = gp( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.04>>> maxima(Integer(0))
0>>> time g = maxima( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.30>>> kash(Integer(0))
0>>> time g = kash( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.04>>> mathematica(Integer(0))
0>>> time g = mathematica( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.03>>> maple(Integer(0))
0>>> time g = maple( 1938^99484 )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.11>>> gap(Integer(0))
0>>> time g = gap.eval( 1938^99484;; )CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 1.02

Note that GAP and Maxima are the slowest in this test (this was run on the machine sage.math.washington.edu).
Because of the pexpect interface overhead, it is perhaps unfair to compare these to Sage, which is the fastest.

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3.5 Other IPython tricks
As noted above, Sage uses IPython as its front end, and so you can use any of IPython’s commands and features. You
can read the full IPython documentation. Meanwhile, here are some fun tricks – these are called “Magic commands” inIPython:• You can use %edit (or %ed or ed) to open an editor, if you want to type in some complex code. Before you
start Sage, make sure that the EDITOR environment variable is set to your favorite editor (by putting export
EDITOR=/usr/bin/emacs or export EDITOR=/usr/bin/vim or something similar in the appropriate
place, like a .profile file). From the Sage prompt, executing %edit will open up the named editor. Then within
the editor you can define a function:
def some_function(n):return n**2 + 3*n + 2

Save and quit from the editor. For the rest of your Sage session, you can then use some_function. If you wantto modify it, type %edit some_function from the Sage prompt.
• If you have a computation and you want to modify its output for another use, perform the computation and type
%rep: this will place the output from the previous command at the Sage prompt, ready for you to edit it.
sage: f(x) = cos(x)sage: f(x).derivative(x)-sin(x)>>> from sage.all import *>>> __tmp__=var("x"); f = symbolic_expression(cos(x)).function(x)>>> f(x).derivative(x)-sin(x)At this point, if you type %rep at the Sage prompt, you will get a new Sage prompt, followed by -sin(x), withthe cursor at the end of the line.
For more, type %quickref to get a quick reference guide to IPython. As of this writing (April 2011), Sage uses version
0.9.1 of IPython, and the documentation for its magic commands is available online. Various slightly advanced aspectsof magic command system are documented here in IPython.

3.6 Errors and Exceptions
When something goes wrong, you will usually see a Python “exception”. Python even tries to suggest what raised theexception. Often you see the name of the exception, e.g., NameError or ValueError (see the Python Library
Reference [PyLR] for a complete list of exceptions). For example,
sage: 3_2-----------------------------------------------------------File "<console>", line 1
ZZ(3)_2^
SyntaxError: invalid ...
sage: EllipticCurve([0,infinity])-----------------------------------------------------------Traceback (most recent call last):...
TypeError: Unable to coerce Infinity (<class sage...Infinity >) to Rational

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Tutorial, Release 10.4>>> from sage.all import *>>> Integer(3_2)-----------------------------------------------------------File "<console>", line 1
ZZ(3)_2^
SyntaxError: invalid ...
>>> EllipticCurve([Integer(0),infinity])-----------------------------------------------------------Traceback (most recent call last):...
TypeError: Unable to coerce Infinity (<class sage...Infinity >) to Rational

The interactive debugger is sometimes useful for understanding what went wrong. You can toggle it on or off using%pdb (the default is off). The prompt ipdb> appears if an exception is raised and the debugger is on. From within the
debugger, you can print the state of any local variable, and move up and down the execution stack. For example,
sage: %pdb
Automatic pdb calling has been turned ON
sage: EllipticCurve([1,infinity])--------------------------------------------------------------------------<class exceptions.TypeError >Traceback (most recent call last)
...
ipdb>
>>> from sage.all import *>>> %pdb
Automatic pdb calling has been turned ON>>> EllipticCurve([Integer(1),infinity])--------------------------------------------------------------------------<class exceptions.TypeError >Traceback (most recent call last)
...
ipdb>

For a list of commands in the debugger, type ? at the ipdb> prompt:ipdb> ?
Documented commands (type help <topic>):========================================
EOF
break commands
debug
h
a
bt
condition disable help
alias ccontdown
ignoreargs
clcontinue
enablejbclear dexit
jump
whatis where

l
list
n
next
p

pdef
pdoc
pinfo
pp
q

quit
r
returns
steptbreakuunaliasupw

Miscellaneous help topics:==========================
exec pdb
Undocumented commands:======================
retval rv

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Type Ctrl-D or quit to return to Sage.

3.7 Reverse Search and Tab Completion
Reverse search: Type the beginning of a command, then Ctrl-p (or just hit the up arrow key) to go back to each line
you have entered that begins in that way. This works even if you completely exit Sage and restart later. You can also do areverse search through the history using Ctrl-r. All these features use the readline package, which is available on
most flavors of Linux.
To illustrate tab completion, first create the three dimensional vector space 𝑉 =3

as follows:

sage: V = VectorSpace(QQ,3)
sage: V
Vector space of dimension 3 over Rational Field>>> from sage.all import *>>> V = VectorSpace(QQ,Integer(3))>>> V
Vector space of dimension 3 over Rational Field

You can also use the following more concise notation:
sage: V = QQ^3>>> from sage.all import *>>> V = QQ**Integer(3)Then it is easy to list all member functions for 𝑉 using tab completion. Just type V., then type the Tab key on yourkeyboard:
sage: V.[tab key]
V._VectorSpace_generic__base_field...
V.ambient_space
V.base_field
V.base_ring
V.basisV.coordinates...
V.zero_vector>>> from sage.all import *>>> V.[tab key]
V._VectorSpace_generic__base_field...
V.ambient_space
V.base_field
V.base_ring
V.basisV.coordinates...
V.zero_vector

If you type the first few letters of a function, then the Tab key, you get only functions that begin as indicated.

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sage: V.i[tab key]
V.is_ambient V.is_dense

V.is_full

V.is_sparse>>> from sage.all import *>>> V.i[tab key]
V.is_ambient V.is_dense

V.is_full

V.is_sparse

If you wonder what a particular function does, e.g., the coordinates function, type V.coordinates? for help or V.
coordinates?? for the source code, as explained in the next section.

3.8 Integrated Help System
Sage features an integrated help facility. Type a function name followed by ? for the documentation for that function.
sage: V = QQ^3sage: V.coordinates?
Type:instancemethod
Base Class:<class instancemethod >String Form:<bound method FreeModule_ambient_field.coordinates of Vector
space of dimension 3 over Rational Field>
Namespace:InteractiveFile:/home/was/s/local/lib/python2.4/site-packages/sage/modules/f
ree_module.py
Definition:
V.coordinates(self, v)
Docstring:Write v in terms of the basis for self.
Returns a list c such that if B is the basis for self, thensum c_i B_i = v.
If v is not in self, raises an ArithmeticError exception.
EXAMPLES:
sage: M = FreeModule(IntegerRing(), 2); M0,M1=M.gens()
sage: W = M.submodule([M0 + M1, M0 - 2*M1])
sage: W.coordinates(2*M0-M1)[2, -1]>>> from sage.all import *>>> V = QQ**Integer(3)>>> V.coordinates?
Type:instancemethod
Base Class:<class instancemethod >String Form:<bound method FreeModule_ambient_field.coordinates of Vector
space of dimension 3 over Rational Field>
Namespace:InteractiveFile:/home/was/s/local/lib/python2.4/site-packages/sage/modules/f
ree_module.py
Definition:
V.coordinates(self, v)
Docstring:Write v in terms of the basis for self.
Returns a list c such that if B is the basis for self, then(continues on next page)98

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(continued from previous page)sum c_i B_i = v.
If v is not in self, raises an ArithmeticError exception.
EXAMPLES:>>> M = FreeModule(IntegerRing(), Integer(2)); M0,M1=M.gens()>>> W = M.submodule([M0 + M1, M0 - Integer(2)*M1])>>> W.coordinates(Integer(2)*M0-M1)[2, -1]As shown above, the output tells you the type of the object, the file in which it is defined, and a useful description of
the function with examples that you can paste into your current session. Almost all of these examples are regularly
automatically tested to make sure they work and behave exactly as claimed.
Another feature that is very much in the spirit of the open source nature of Sage is that if f is a Python function, thentyping f?? displays the source code that defines f. For example,
sage: V = QQ^3sage: V.coordinates??
Type:instancemethod...
Source:def coordinates(self, v):"""
Write $v$ in terms of the basis for self.
...
"""
return self.coordinate_vector(v).list()>>> from sage.all import *>>> V = QQ**Integer(3)>>> V.coordinates??
Type:instancemethod...
Source:def coordinates(self, v):"""
Write $v$ in terms of the basis for self.
...
"""
return self.coordinate_vector(v).list()

This tells us that all the coordinates function does is call the coordinate_vector function and change the result
into a list. What does the coordinate_vector function do?
sage: V = QQ^3sage: V.coordinate_vector??
...
def coordinate_vector(self, v):...
return self.ambient_vector_space()(v)>>> from sage.all import *>>> V = QQ**Integer(3)>>> V.coordinate_vector??
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(continued from previous page)...
def coordinate_vector(self, v):...
return self.ambient_vector_space()(v)

The coordinate_vector function coerces its input into the ambient space, which has the effect of computing thevector of coefficients of 𝑣 in terms of 𝑉 . The space 𝑉 is already ambient since it’s just 3 . There is also a coordinate_vector function for subspaces, and it’s different. We create a subspace and see:sage: V = QQ^3; W = V.span_of_basis([V.0, V.1])
sage: W.coordinate_vector??
...
def coordinate_vector(self, v):
"""
...
"""
# First find the coordinates of v wrt echelon basis.
w = self.echelon_coordinate_vector(v)# Next use transformation matrix from echelon basis to# user basis.
T = self.echelon_to_user_matrix()
return T.linear_combination_of_rows(w)>>> from sage.all import *>>> V = QQ**Integer(3); W = V.span_of_basis([V.gen(0), V.gen(1)])>>> W.coordinate_vector??
...
def coordinate_vector(self, v):
"""
...
"""
# First find the coordinates of v wrt echelon basis.
w = self.echelon_coordinate_vector(v)# Next use transformation matrix from echelon basis to# user basis.
T = self.echelon_to_user_matrix()
return T.linear_combination_of_rows(w)(If you think the implementation is inefficient, please sign up to help optimize linear algebra.)
You may also type help(command_name) or help(class) for a manpage-like help file about a given class.
sage: help(VectorSpace)
Help on function VectorSpace in module sage.modules.free_module:
VectorSpace(K, dimension_or_basis_keys=None, sparse=False, inner_product_matrix=None,␣˓→*,
with_basis= standard , dimension=None, basis_keys=None, **args)
EXAMPLES:
The base can be complicated, as long as it is a field.
::
sage: V = VectorSpace(FractionField(PolynomialRing(ZZ, x )),3)
sage: V
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(continued from previous page)

Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in x
over Integer Ringsage: V.basis()[(1, 0, 0),(0, 1, 0),--More->>> from sage.all import *>>> help(VectorSpace)
Help on function VectorSpace in module sage.modules.free_module:
VectorSpace(K, dimension_or_basis_keys=None, sparse=False, inner_product_matrix=None,␣˓→*,
with_basis= standard , dimension=None, basis_keys=None, **args)
EXAMPLES:
The base can be complicated, as long as it is a field.
::>>> V = VectorSpace(FractionField(PolynomialRing(ZZ, x )),Integer(3))>>> V
Vector space of dimension 3 over Fraction Field of Univariate Polynomial Ring in x
over Integer Ring>>> V.basis()[(1, 0, 0),(0, 1, 0),--More--When you type q to exit the help system, your session appears just as it was. The help listing does not clutter up
your session, unlike the output of function_name? sometimes does. It’s particularly helpful to type help(module_name). For example, vector spaces are defined in sage.modules.free_module, so type help(sage.
modules.free_module) for documentation about that whole module. When viewing documentation using help,
you can search by typing / and in reverse by typing ?.

3.9 Saving and Loading Individual Objects
Suppose you compute a matrix or worse, a complicated space of modular symbols, and would like to save it for later use.
What can you do? There are several approaches that computer algebra systems take to saving individual objects.
1. Save your Game: Only support saving and loading of complete sessions (e.g., GAP, Magma).
2. Unified Input/Output: Make every object print in a way that can be read back in (GP/PARI).
3. Eval: Make it easy to evaluate arbitrary code in the interpreter (e.g., Singular, PARI).
Because Sage uses Python, it takes a different approach, which is that every object can be serialized, i.e., turned into astring from which that object can be recovered. This is in spirit similar to the unified I/O approach of PARI, except it
doesn’t have the drawback that objects print to screen in too complicated of a way. Also, support for saving and loading is(in most cases) completely automatic, requiring no extra programming; it’s simply a feature of Python that was designed
into the language from the ground up.
Almost all Sage objects x can be saved in compressed form to disk using save(x, filename) (or in many cases
x.save(filename)). To load the object back in, use load(filename).
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sage: A = MatrixSpace(QQ,3)(range(9))^2
sage: A
[ 15 18 21][ 42 54 66][ 69 90 111]
sage: save(A, A )>>> from sage.all import *>>> A = MatrixSpace(QQ,Integer(3))(range(Integer(9)))**Integer(2)>>> A[ 15 18 21][ 42 54 66][ 69 90 111]>>> save(A, A )You should now quit Sage and restart. Then you can get A back:sage: A = load( A )sage: A
[ 15 18 21][ 42 54 66][ 69 90 111]>>> from sage.all import *>>> A = load( A )>>> A[ 15 18 21][ 42 54 66][ 69 90 111]

You can do the same with more complicated objects, e.g., elliptic curves. All data about the object that is cached is storedwith the object. For example,
sage: E = EllipticCurve( 11a )sage: v = E.anlist(100000)
sage: save(E, E )sage: quit>>> from sage.all import *>>> E = EllipticCurve( 11a )>>> v = E.anlist(Integer(100000))>>> save(E, E )>>> quit# takes a while# takes a while

The saved version of E takes 153 kilobytes, since it stores the first 100000 𝑎𝑛 with it.
~/tmp$ ls -l E.sobj-rw-r--r-- 1 was was 153500 2006-01-28 19:23 E.sobj~/tmp$ sage [...]
sage: E = load( E )sage: v = E.anlist(100000)# instant!

(In Python, saving and loading is accomplished using the cPickle module. In particular, a Sage object x can be saved
via cPickle.dumps(x, 2). Note the 2!)
Sage cannot save and load individual objects created in some other computer algebra systems, e.g., GAP, Singular, Maxima, etc. They reload in a state marked “invalid”. In GAP, though many objects print in a form from which they can be
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reconstructed, many don’t, so reconstructing from their print representation is purposely not allowed.
sage: a = gap(2)sage: a.save( a )sage: load( a )Traceback (most recent call last):...
ValueError: The session in which this object was defined is no longer
running.
>>> from sage.all import *>>> a = gap(Integer(2))>>> a.save( a )>>> load( a )Traceback (most recent call last):...
ValueError: The session in which this object was defined is no longer
running.

GP/PARI objects can be saved and loaded since their print representation is enough to reconstruct them.
sage: a = gp(2)sage: a.save( a )sage: load( a )2>>> from sage.all import *>>> a = gp(Integer(2))>>> a.save( a )>>> load( a )2

Saved objects can be re-loaded later on computers with different architectures or operating systems, e.g., you could save ahuge matrix on 32-bit OS X and reload it on 64-bit Linux, find the echelon form, then move it back. Also, in many cases
you can even load objects into versions of Sage that are different than the versions they were saved in, as long as the codefor that object isn’t too different. All the attributes of the objects are saved, along with the class (but not source code) thatdefines the object. If that class no longer exists in a new version of Sage, then the object can’t be reloaded in that newer
version. But you could load it in an old version, get the objects dictionary (with x.__dict__), and save the dictionary,and load that into the newer version.

3.9.1 Saving as Text
You can also save the ASCII text representation of objects to a plain text file by simply opening a file in write mode andwriting the string representation of the object (you can write many objects this way as well). When you’re done writing
objects, close the file.
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: f = (x+y)^7
sage: o = open( file.txt , w )sage: o.write(str(f))
sage: o.close()>>> from sage.all import *>>> R = PolynomialRing(QQ,Integer(2), names=( x ,>>> f = (x+y)**Integer(7)y ,)); (x, y,) = R._first_ngens(2)(continues on next page)

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(continued from previous page)>>> o = open( file.txt , w )>>> o.write(str(f))>>> o.close()

3.10 Saving and Loading Complete Sessions
Sage has very flexible support for saving and loading complete sessions.
The command save_session(sessionname) saves all the variables you’ve defined in the current session as adictionary in the given sessionname. (In the rare case when a variable does not support saving, it is simply not saved
to the dictionary.) The resulting file is an .sobj file and can be loaded just like any other object that was saved. When
you load the objects saved in a session, you get a dictionary whose keys are the variables names and whose values are theobjects.
You can use the load_session(sessionname) command to load the variables defined in sessionname into
the current session. Note that this does not wipe out variables you’ve already defined in your current session; instead, thetwo sessions are merged.
First we start Sage and define some variables.
sage: E = EllipticCurve( 11a )sage: M = ModularSymbols(37)sage: a = 389
sage: t = M.T(2003).matrix(); t.charpoly().factor()_4 = (x - 2004) * (x - 12)^2 * (x + 54)^2>>> from sage.all import *>>> E = EllipticCurve( 11a )>>> M = ModularSymbols(Integer(37))>>> a = Integer(389)>>> t = M.T(Integer(2003)).matrix(); t.charpoly().factor()_4 = (x - 2004) * (x - 12)^2 * (x + 54)^2

Next we save our session, which saves each of the above variables into a file. Then we view the file, which is about 3K insize.
sage: save_session( misc )Saving a
Saving M
Saving t
Saving E
sage: quit
was@form:~/tmp$ ls -l misc.sobj-rw-r--r-- 1 was was 2979 2006-01-28 19:47 misc.sobj>>> from sage.all import *>>> save_session( misc )Saving a
Saving M
Saving t
Saving E>>> quit
was@form:~/tmp$ ls -l misc.sobj-rw-r--r-- 1 was was 2979 2006-01-28 19:47 misc.sobj

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Finally we restart Sage, define an extra variable, and load our saved session.
sage: b = 19
sage: load_session( misc )Loading a
Loading M
Loading E
Loading t>>> from sage.all import *>>> b = Integer(19)>>> load_session( misc )Loading a
Loading M
Loading E
Loading t

Each saved variable is again available. Moreover, the variable b was not overwritten.
sage: M
Full Modular Symbols space for Gamma_0(37) of weight 2 with sign 0
and dimension 5 over Rational Field
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational
Field
sage: b19
sage: a389>>> from sage.all import *>>> M
Full Modular Symbols space for Gamma_0(37) of weight 2 with sign 0
and dimension 5 over Rational Field>>> E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational
Field
>>> b
19
>>> a
389

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CHAPTER

FOUR

INTERFACES

A central facet of Sage is that it supports computation with objects in many different computer algebra systems “under
one roof” using a common interface and clean programming language.
The console and interact methods of an interface do very different things. For example, using GAP as an example:
1. gap.console(): This opens the GAP console - it transfers control to GAP. Here Sage is serving as nothing
more than a convenient program launcher, similar to the Linux bash shell.
2. gap.interact(): This is a convenient way to interact with a running GAP instance that may be “full of” Sage
objects. You can import Sage objects into this GAP session (even from the interactive interface), etc.

4.1 GP/PARIPARI is a compact, very mature, highly optimized C program whose primary focus is number theory. There are two very
distinct interfaces that you can use in Sage:• gp – the “G o P ARI” interpreter, and• pari – the PARI C library.
For example, the following are two ways of doing the same thing. They look identical, but the output is actually different,
and what happens behind the scenes is drastically different.
sage: gp( znprimroot(10007) )Mod(5, 10007)
sage: pari( znprimroot(10007) )Mod(5, 10007)>>> from sage.all import *>>> gp( znprimroot(10007) )Mod(5, 10007)>>> pari( znprimroot(10007) )Mod(5, 10007)

In the first case, a separate copy of the GP interpreter is started as a server, and the string znprimroot(10007) issent to it, evaluated by GP, and the result is assigned to a variable in GP (which takes up space in the child GP processes
memory that won’t be freed). Then the value of that variable is displayed. In the second case, no separate program isstarted, and the string znprimroot(10007) is evaluated by a certain PARI C library function. The result is stored
in a piece of memory on the Python heap, which is freed when the variable is no longer referenced. The objects havedifferent types:

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sage: type(gp( znprimroot(10007) ))<class sage.interfaces.gp.GpElement >sage: type(pari( znprimroot(10007) ))<class cypari2.gen.Gen >>>> from sage.all import *>>> type(gp( znprimroot(10007) ))<class sage.interfaces.gp.GpElement >>>> type(pari( znprimroot(10007) ))<class cypari2.gen.Gen >So which should you use? It depends on what you’re doing. The GP interface can do absolutely anything you could doin the usual GP/PARI command line program, since it is running that program. In particular, you can load complicated
PARI programs and run them. In contrast, the PARI interface (via the C library) is much more restrictive. First, not all
member functions have been implemented. Second, a lot of code, e.g., involving numerical integration, won’t work via
the PARI interface. That said, the PARI interface can be significantly faster and more robust than the GP one.
(If the GP interface runs out of memory evaluating a given input line, it will silently and automatically double the stacksize and retry that input line. Thus your computation won’t crash if you didn’t correctly anticipate the amount of memorythat would be needed. This is a nice trick the usual GP interpreter doesn’t seem to provide. Regarding the PARI C libraryinterface, it immediately copies each created object off of the PARI stack, hence the stack never grows. However, eachobject must not exceed 100MB in size, or the stack will overflow when the object is being created. This extra copying
does impose a slight performance penalty.)
In summary, Sage uses the PARI C library to provide functionality similar to that provided by the GP/PARI interpreter,
except with different sophisticated memory management and the Python programming language.
First we create a PARI list from a Python list.
sage: v = pari([1,2,3,4,5])
sage: v
[1, 2, 3, 4, 5]
sage: type(v)<class cypari2.gen.Gen >>>> from sage.all import *>>> v = pari([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])>>> v[1, 2, 3, 4, 5]>>> type(v)<class cypari2.gen.Gen >Every PARI object is of type Gen. The PARI type of the underlying object can be obtained using the type member
function.
sage: v.type()
t_VEC>>> from sage.all import *>>> v.type()
t_VEC

In PARI, to create an elliptic curve we enter ellinit([1,2,3,4,5]). Sage is similar, except that ellinit is amethod that can be called on any PARI object, e.g., our t_VEC 𝑣.

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sage: e = v.ellinit()
sage: e.type()t_VEC
sage: pari(e)[:13][1, 2, 3, 4, 5, 9, 11, 29, 35, -183, -3429, -10351, 6128487/10351]>>> from sage.all import *>>> e = v.ellinit()>>> e.type()t_VEC>>> pari(e)[:Integer(13)][1, 2, 3, 4, 5, 9, 11, 29, 35, -183, -3429, -10351, 6128487/10351]

Now that we have an elliptic curve object, we can compute some things about it.
sage: e.elltors()[1, [], []]sage: e.ellglobalred()[10351, [1, -1, 0, -1], 1, [11, 1; 941, 1], [[1, 5, 0, 1], [1, 5, 0, 1]]]
sage: f = e.ellchangecurve([1,-1,0,-1])
sage: f[:5][1, -1, 0, 4, 3]>>> from sage.all import *>>> e.elltors()[1, [], []]>>> e.ellglobalred()[10351, [1, -1, 0, -1], 1, [11, 1; 941, 1], [[1, 5, 0, 1], [1, 5, 0, 1]]]>>> f = e.ellchangecurve([Integer(1),-Integer(1),Integer(0),-Integer(1)])>>> f[:Integer(5)][1, -1, 0, 4, 3]

4.2 GAP
Sage comes with GAP for computational discrete mathematics, especially group theory.
Here’s an example of GAP’s IdGroup function.
sage: G = gap( Group((1,2,3)(4,5), (3,4)) )sage: GGroup( [ (1,2,3)(4,5), (3,4) ] )sage: G.Center()Group( () )sage: G.IdGroup()[ 120, 34 ]sage: G.Order()120>>> from sage.all import *>>> G = gap( Group((1,2,3)(4,5), (3,4)) )>>> G
Group( [ (1,2,3)(4,5), (3,4) ] )>>> G.Center()Group( () )(continues on next page)

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(continued from previous page)>>> G.IdGroup()[ 120, 34 ]>>> G.Order()120

We can do the same computation in Sage without explicitly invoking the GAP interface as follows:
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])sage: G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,˓→5)])
sage: G.group_id()[120, 34]
sage: n = G.order(); n120>>> from sage.all import *>>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),˓→Integer(5))],[(Integer(3),Integer(4))]])>>> G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,˓→5)])>>> G.group_id()[120, 34]>>> n = G.order(); n120

For some GAP functionality, you should install an optional Sage packages. This can be done with the command:
sage -i gap_packages

4.3 Singular
Singular provides a massive and mature library for Gröbner bases, multivariate polynomial gcds, bases of Riemann-Rochspaces of a plane curve, and factorizations, among other things. We illustrate multivariate polynomial factorization using
the Sage interface to Singular (do not type the ....:):
sage: R1 = singular.ring(0, (x,y) , dp )sage: R1
polynomial ring, over a field, global ordering//
coefficients: QQ//
number of vars : 2//
block
1 : ordering dp//: names
x y
//
block
2 : ordering C
sage: f = singular( 9*y^8 - 9*x^2*y^7 - 18*x^3*y^6 - 18*x^5*y^6 +....:9*x^6*y^4 + 18*x^7*y^5 + 36*x^8*y^4 + 9*x^10*y^4 - 18*x^11*y^2 ....:
9*x^12*y^3 - 18*x^13*y^2 + 9*x^16 )>>> from sage.all import *>>> R1 = singular.ring(Integer(0), (x,y) , dp )>>> R1polynomial ring, over a field, global ordering(continues on next page)

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(continued from previous page)//coefficients: QQ//
number of vars : 2//
block
1 : ordering dp//: names
x y
//
block
2 : ordering C>>> f = singular( 9*y^8 - 9*x^2*y^7 - 18*x^3*y^6 - 18*x^5*y^6 +...
9*x^6*y^4 + 18*x^7*y^5 + 36*x^8*y^4 + 9*x^10*y^4 - 18*x^11*y^2 ...
9*x^12*y^3 - 18*x^13*y^2 + 9*x^16 )Now that we have defined 𝑓 , we print it and factor.
sage: f9*x^16-18*x^13*y^2-9*x^12*y^3+9*x^10*y^4-18*x^11*y^2+36*x^8*y^4+18*x^7*y^5-18*x^5*y^
˓→6+9*x^6*y^4-18*x^3*y^6-9*x^2*y^7+9*y^8
sage: f.parent()
Singularsage: F = f.factorize(); F[1]:_[1]=9_[2]=x^6-2*x^3*y^2-x^2*y^3+y^4_[3]=-x^5+y^2
[2]:
1,1,2
sage: F[1][2]x^6-2*x^3*y^2-x^2*y^3+y^4>>> from sage.all import *>>> f
9*x^16-18*x^13*y^2-9*x^12*y^3+9*x^10*y^4-18*x^11*y^2+36*x^8*y^4+18*x^7*y^5-18*x^5*y^
˓→6+9*x^6*y^4-18*x^3*y^6-9*x^2*y^7+9*y^8>>> f.parent()
Singular>>> F = f.factorize(); F[1]:_[1]=9_[2]=x^6-2*x^3*y^2-x^2*y^3+y^4_[3]=-x^5+y^2
[2]:
1,1,2
>>> F[Integer(1)][Integer(2)]x^6-2*x^3*y^2-x^2*y^3+y^4

As with the GAP example in GAP, we can compute the above factorization without explicitly using the Singular interface(however, behind the scenes Sage uses the Singular interface for the actual computation). Do not type the ....::
sage: x, y = QQ[ x, y ].gens()
sage: f = (9*y^8 - 9*x^2*y^7 - 18*x^3*y^6 - 18*x^5*y^6 + 9*x^6*y^4....:+ 18*x^7*y^5 + 36*x^8*y^4 + 9*x^10*y^4 - 18*x^11*y^2 - 9*x^12*y^3....:- 18*x^13*y^2 + 9*x^16)
sage: factor(f)(9) * (-x^5 + y^2)^2 * (x^6 - 2*x^3*y^2 - x^2*y^3 + y^4)>>> from sage.all import *>>> x, y = QQ[ x, y ].gens()>>> f = (Integer(9)*y**Integer(8) - Integer(9)*x**Integer(2)*y**Integer(7) -␣˓→Integer(18)*x**Integer(3)*y**Integer(6) - Integer(18)*x**Integer(5)*y**Integer(6) +␣
(continues on next page)

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(continued from previous page)

Integer(9)*x**Integer(6)*y**Integer(4)...
+ Integer(18)*x**Integer(7)*y**Integer(5) +␣˓→Integer(36)*x**Integer(8)*y**Integer(4) + Integer(9)*x**Integer(10)*y**Integer(4) -␣˓→Integer(18)*x**Integer(11)*y**Integer(2) - Integer(9)*x**Integer(12)*y**Integer(3)...
- Integer(18)*x**Integer(13)*y**Integer(2) + Integer(9)*x**Integer(16))>>> factor(f)(9) * (-x^5 + y^2)^2 * (x^6 - 2*x^3*y^2 - x^2*y^3 + y^4)˓→

4.4 Maxima
Maxima is included with Sage, as well as a Lisp implementation. The gnuplot package (which Maxima uses by default for
plotting) is distributed as a Sage optional package. Among other things, Maxima does symbolic manipulation. Maxima
can integrate and differentiate functions symbolically, solve 1st order ODEs, most linear 2nd order ODEs, and has implemented the Laplace transform method for linear ODEs of any degree. Maxima also knows about a wide range of special
functions, has plotting capabilities via gnuplot, and has methods to solve and manipulate matrices (such as row reduction,
eigenvalues and eigenvectors), and polynomial equations.
We illustrate the Sage/Maxima interface by constructing the matrix whose 𝑖, 𝑗 entry is 𝑖/𝑗, for 𝑖, 𝑗 = 1, . . . , 4.
sage: f = maxima.eval( ij_entry[i,j] := i/j )sage: A = maxima( genmatrix(ij_entry,4,4) ); A
matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1])
sage: A.determinant()
0sage: A.echelon()
matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0])
sage: A.eigenvalues()[[0,4],[3,1]]
sage: A.eigenvectors().sage()[[[0, 4], [3, 1]], [[[1, 0, 0, -4], [0, 1, 0, -2], [0, 0, 1, -4/3]], [[1, 2, 3, 4]]]]>>> from sage.all import *>>> f = maxima.eval( ij_entry[i,j] := i/j )>>> A = maxima( genmatrix(ij_entry,4,4) ); A
matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1])>>> A.determinant()
0>>> A.echelon()
matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0])>>> A.eigenvalues()[[0,4],[3,1]]>>> A.eigenvectors().sage()[[[0, 4], [3, 1]], [[[1, 0, 0, -4], [0, 1, 0, -2], [0, 0, 1, -4/3]], [[1, 2, 3, 4]]]]Here’s another example:
sage: A = maxima("matrix ([1, 0, 0], [1, -1, 0], [1, 3, -2])")sage: eigA = A.eigenvectors()
sage: V = VectorSpace(QQ,3)
sage: eigA[[[-2,-1,1],[1,1,1]],[[[0,0,1]],[[0,1,3]],[[1,1/2,5/6]]]]sage: v1 = V(sage_eval(repr(eigA[1][0][0]))); lambda1 = eigA[0][0][0]sage: v2 = V(sage_eval(repr(eigA[1][1][0]))); lambda2 = eigA[0][0][1](continues on next page)

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sage: v3 = V(sage_eval(repr(eigA[1][2][0]))); lambda3 = eigA[0][0][2]sage: M = MatrixSpace(QQ,3,3)
sage: AA = M([[1,0,0],[1, - 1,0],[1,3, - 2]])sage: b1 = v1.base_ring()sage: AA*v1 == b1(lambda1)*v1
True
sage: b2 = v2.base_ring()sage: AA*v2 == b2(lambda2)*v2
True
sage: b3 = v3.base_ring()sage: AA*v3 == b3(lambda3)*v3
True>>> from sage.all import *>>> A = maxima("matrix ([1, 0, 0], [1, -1, 0], [1, 3, -2])")>>> eigA = A.eigenvectors()>>> V = VectorSpace(QQ,Integer(3))>>> eigA[[[-2,-1,1],[1,1,1]],[[[0,0,1]],[[0,1,3]],[[1,1/2,5/6]]]]>>> v1 = V(sage_eval(repr(eigA[Integer(1)][Integer(0)][Integer(0)]))); lambda1 =␣
˓→eigA[Integer(0)][Integer(0)][Integer(0)]>>> v2 = V(sage_eval(repr(eigA[Integer(1)][Integer(1)][Integer(0)]))); lambda2 =␣˓→eigA[Integer(0)][Integer(0)][Integer(1)]>>> v3 = V(sage_eval(repr(eigA[Integer(1)][Integer(2)][Integer(0)]))); lambda3 =␣
˓→eigA[Integer(0)][Integer(0)][Integer(2)]>>> M = MatrixSpace(QQ,Integer(3),Integer(3))>>> AA = M([[Integer(1),Integer(0),Integer(0)],[Integer(1), - Integer(1),Integer(0)],˓→[Integer(1),Integer(3), - Integer(2)]])>>> b1 = v1.base_ring()>>> AA*v1 == b1(lambda1)*v1
True>>> b2 = v2.base_ring()>>> AA*v2 == b2(lambda2)*v2
True>>> b3 = v3.base_ring()>>> AA*v3 == b3(lambda3)*v3
True

Finally, we give an example of using Sage to plot using openmath. Many of these were modified from the Maxima
reference manual.
A 2D plot of several functions (do not type the ....:):
sage: maxima.plot2d( [cos(7*x),cos(23*x)^4,sin(13*x)^3] , [x,0,1] ,....:[plot_format,openmath] )>>> from sage.all import *>>> maxima.plot2d( [cos(7*x),cos(23*x)^4,sin(13*x)^3] , [x,0,1] ,...
[plot_format,openmath] )# not tested# not tested

A “live” 3D plot which you can move with your mouse (do not type the ....:):
sage: maxima.plot3d ("2^(-u^2 + v^2)", "[u, -3, 3]", "[v, -2, 2]",....:[plot_format, openmath] )# not tested(continues on next page)

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sage: maxima.plot3d("atan(-x^2 + y^3/4)", "[x, -4, 4]", "[y, -4, 4]",....:"[grid, 50, 50]", [plot_format, openmath] )# not tested>>> from sage.all import *>>> maxima.plot3d ("2^(-u^2 + v^2)", "[u, -3, 3]", "[v, -2, 2]", # not tested...
[plot_format, openmath] )>>> maxima.plot3d("atan(-x^2 + y^3/4)", "[x, -4, 4]", "[y, -4, 4]", # not tested...
"[grid, 50, 50]", [plot_format, openmath] )The next plot is the famous Möbius strip (do not type the ....:):
sage: maxima.plot3d("[cos(x)*(3 + y*cos(x/2)), sin(x)*(3 + y*cos(x/2)), y*sin(x/2)]",␣
˓→ # not tested....:"[x, -4, 4]", "[y, -4, 4]", [plot_format, openmath] )>>> from sage.all import *>>> maxima.plot3d("[cos(x)*(3 + y*cos(x/2)), sin(x)*(3 + y*cos(x/2)), y*sin(x/2)]",
˓→# not tested...
"[x, -4, 4]", "[y, -4, 4]", [plot_format, openmath] )The next plot is the famous Klein bottle (do not type the ....:):
sage: maxima("expr_1: 5*cos(x)*(cos(x/2)*cos(y) + sin(x/2)*sin(2*y)+ 3.0) - 10.0")
5*cos(x)*(sin(x/2)*sin(2*y)+cos(x/2)*cos(y)+3.0)-10.0
sage: maxima("expr_2: -5*sin(x)*(cos(x/2)*cos(y) + sin(x/2)*sin(2*y)+ 3.0)").sage()-5*(cos(1/2*x)*cos(y) + sin(1/2*x)*sin(2*y) + 3.0)*sin(x)sage: maxima("expr_3: 5*(-sin(x/2)*cos(y) + cos(x/2)*sin(2*y))")5*(cos(x/2)*sin(2*y)-sin(x/2)*cos(y))
sage: maxima.plot3d ("[expr_1, expr_2, expr_3]", "[x, -%pi, %pi]", # not tested....:"[y, -%pi, %pi]", "[ grid, 40, 40]", [plot_format, openmath] )>>> from sage.all import *>>> maxima("expr_1: 5*cos(x)*(cos(x/2)*cos(y) + sin(x/2)*sin(2*y)+ 3.0) - 10.0")
5*cos(x)*(sin(x/2)*sin(2*y)+cos(x/2)*cos(y)+3.0)-10.0>>> maxima("expr_2: -5*sin(x)*(cos(x/2)*cos(y) + sin(x/2)*sin(2*y)+ 3.0)").sage()-5*(cos(1/2*x)*cos(y) + sin(1/2*x)*sin(2*y) + 3.0)*sin(x)>>> maxima("expr_3: 5*(-sin(x/2)*cos(y) + cos(x/2)*sin(2*y))")5*(cos(x/2)*sin(2*y)-sin(x/2)*cos(y))>>> maxima.plot3d ("[expr_1, expr_2, expr_3]", "[x, -%pi, %pi]", # not tested...
"[y, -%pi, %pi]", "[ grid, 40, 40]", [plot_format, openmath] )114

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FIVE

SAGE, LATEX AND FRIENDS

Sage and the LaTeX dialect of TeX have an intensely synergistic relationship. This section aims to introduce the variety
of interactions, beginning with the most basic and proceeding to the more unusual.

5.1 Basic Use
Every “object” in Sage is required to have a LaTeX representation. You can access this representation by executing
latex(foo) where foo is some object in Sage. The output is a string that should render a reasonably accurate representation of foo when used in TeX’s math-mode (for example, when enclosed between a pair of single dollar signs).
Some examples of this follow below.
sage: var( z )zsage: latex(z^12)z^{12}
sage: latex(sqrt(z^2 + 1/2))\sqrt{z^{2} + \frac{1}{2}}sage: latex( a string )\text{\texttt{a{ }string}}sage: latex(QQ)\Bold{Q}sage: latex(ZZ[ x ])\Bold{Z}[x]sage: latex(matrix(QQ, 2, 3, [[2,4,6],[-1,-1,-1]]))\left(\begin{array}{rrr}
2 & 4 & 6 \\-1 & -1 & -1\end{array}\right)>>> from sage.all import *>>> var( z )z>>> latex(z**Integer(12))
z^{12}>>> latex(sqrt(z**Integer(2) + Integer(1)/Integer(2)))\sqrt{z^{2} + \frac{1}{2}}>>> latex( a string )\text{\texttt{a{ }string}}>>> latex(QQ)\Bold{Q}>>> latex(ZZ[ x ])\Bold{Z}[x](continues on next page)115

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(continued from previous page)>>> latex(matrix(QQ, Integer(2), Integer(3), [[Integer(2),Integer(4),Integer(6)],[˓→Integer(1),-Integer(1),-Integer(1)]]))\left(\begin{array}{rrr}
2 & 4 & 6 \\-1 & -1 & -1\end{array}\right)

In this way, Sage can be used effectively for constructing portions of a LaTeX document: create or compute an object inSage, do latex(foo) of the object foo and cut/paste the LaTeX string into your document.
The command view(foo) will show the rendered LaTeX representation of foo. In the background, the commandruns latex(foo) and incorporates the LaTeX string into a simple LaTeX document, processes that document withthe system-wide TeX installation, and finally an appropriate viewer will be called to display the output.
In the Jupyter notebook, you can see the rendered LaTeX representation of the output of the entered commands automatically. You can start this automatic rendering by executing %display latex (and stop by executing %display
plain).
The Jupyter notebook uses MathJax to render mathematics cleanly in a web browser. MathJax is an open source JavaScript
display engine for mathematics that works in all modern browsers. It is able to render a large, but not totally complete,
subset of LaTeX. It has no support for things like complicated tables, sectioning or document management, as it is oriented
towards accurately rendering math snippets of LaTeX.
The automatic LaTeX rendering in the Jupyter notebook (with %display latex on) is internally done via the sage.
misc.html.MathJax class. The object of this class converts a Sage object through latex() to a form of HTMLpalatable to MathJax and then wraps it in HTML.
sage: from sage.misc.html import MathJax
sage: mj = MathJax()
sage: var( z )zsage: mj(z^12)<html>\[z^{12}\]</html>
sage: mj(sqrt(z^2 + 1/2))<html>\[\sqrt{z^{2} + \frac{1}{2}}\]</html>
sage: mj( a string )<html>\[\verb|a|\verb| |\verb|string|\]</html>
sage: mj(QQ)<html>\[\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}\]</html>
sage: mj(ZZ[ x ])<html>\[\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Z}[x]\]</html>
sage: mj(matrix(QQ, 2, 3, [[2,4,6],[-1,-1,-1]]))<html>\[\left(\begin{array}{rrr}
2 & 4 & 6 \\-1 & -1 & -1\end{array}\right)\]</html>>>> from sage.all import *>>> from sage.misc.html import MathJax>>> mj = MathJax()>>> var( z )z>>> mj(z**Integer(12))<html>\[z^{12}\]</html>>>> mj(sqrt(z**Integer(2) + Integer(1)/Integer(2)))<html>\[\sqrt{z^{2} + \frac{1}{2}}\]</html>>>> mj( a string )(continues on next page)

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(continued from previous page)<html>\[\verb|a|\verb| |\verb|string|\]</html>>>> mj(QQ)<html>\[\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}\]</html>>>> mj(ZZ[ x ])<html>\[\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Z}[x]\]</html>>>> mj(matrix(QQ, Integer(2), Integer(3), [[Integer(2),Integer(4),Integer(6)],[˓→Integer(1),-Integer(1),-Integer(1)]]))<html>\[\left(\begin{array}{rrr}
2 & 4 & 6 \\-1 & -1 & -1\end{array}\right)\]</html>

This is useful to know if you need to understand the LaTeX rendering of a Sage object.

5.2 Customizing LaTeX Generation
There are several ways to customize the actual LaTeX code generated by the latex() command. There is a pre-defined
object named latex which has several methods, which you can list by typing latex., followed by the Tab key (notethe period).
A good example is the latex.matrix_delimiters method. It can be used to change the notation surrounding
a matrix – large parentheses, brackets, braces, vertical bars. No notion of style is enforced, you can mix and match as
you please. Notice how the backslashes needed in LaTeX require an extra slash so they are escaped properly within the
Python string.
sage: A = matrix(ZZ, 2, 2, range(4))
sage: latex(A)\left(\begin{array}{rr}0 & 1 \\
2 & 3\end{array}\right)
sage: latex.matrix_delimiters(left= [ , right= ] )sage: latex(A)\left[\begin{array}{rr}0 & 1 \\
2 & 3\end{array}\right]
sage: latex.matrix_delimiters(left= \\{ , right= \\} )sage: latex(A)\left\{\begin{array}{rr}0 & 1 \\
2 & 3\end{array}\right\}>>> from sage.all import *>>> A = matrix(ZZ, Integer(2), Integer(2), range(Integer(4)))>>> latex(A)\left(\begin{array}{rr}0 & 1 \\
2 & 3\end{array}\right)>>> latex.matrix_delimiters(left= [ , right= ] )>>> latex(A)\left[\begin{array}{rr}(continues on next page)

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0 & 1 \\
2 & 3\end{array}\right]>>> latex.matrix_delimiters(left= \\{ , right= \\} )>>> latex(A)\left\{\begin{array}{rr}0 & 1 \\
2 & 3\end{array}\right\}

The latex.vector_delimiters method works similarly.
The way common rings and fields (integers, rational, reals, etc.) are typeset can be controlled by the latex.
blackboard_bold method. These sets are by default typeset in bold, but may optionally be written in a double-struck
fashion as sometimes done in written work. This is accomplished by redefining the \Bold{} macro which is built-in toSage.
sage: latex(QQ)\Bold{Q}sage: from sage.misc.html import MathJax
sage: mj = MathJax()
sage: mj(QQ)<html>\[\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}\]</html>
sage: latex.blackboard_bold(True)
sage: mj(QQ)<html>\[\newcommand{\Bold}[1]{\mathbb{#1}}\Bold{Q}\]</html>
sage: latex.blackboard_bold(False)>>> from sage.all import *>>> latex(QQ)\Bold{Q}>>> from sage.misc.html import MathJax>>> mj = MathJax()>>> mj(QQ)<html>\[\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}\]</html>>>> latex.blackboard_bold(True)>>> mj(QQ)<html>\[\newcommand{\Bold}[1]{\mathbb{#1}}\Bold{Q}\]</html>>>> latex.blackboard_bold(False)

It is possible to take advantage of the extensible nature of LaTeX by adding in new macros. Individual macros can be
added so that they are used when MathJax interprets a LaTeX snippet.
sage: latex.add_macro(r"\newcommand{\sqrt}[1]{(#1)^\frac{1}{2}}")sage: latex.extra_macros()\\newcommand{\\sqrt}[1]{(#1)^\\frac{1}{2}}sage: var( x y )(x, y)
sage: latex(sqrt(x+y))\sqrt{x + y}sage: from sage.misc.html import MathJax
sage: mj = MathJax()
sage: mj(sqrt(x + y))<html>\[\newcommand{\sqrt}[1]{(#1)^\frac{1}{2}}\sqrt{x + y}\]</html>
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Tutorial, Release 10.4>>> from sage.all import *>>> latex.add_macro(r"\newcommand{\sqrt}[1]{(#1)^\frac{1}{2}}")>>> latex.extra_macros()\\newcommand{\\sqrt}[1]{(#1)^\\frac{1}{2}}>>> var( x y )(x, y)>>> latex(sqrt(x+y))\sqrt{x + y}>>> from sage.misc.html import MathJax>>> mj = MathJax()>>> mj(sqrt(x + y))<html>\[\newcommand{\sqrt}[1]{(#1)^\frac{1}{2}}\sqrt{x + y}\]</html>>>> latex.extra_macros( )5.3 Customizing LaTeX Processing
The system-wide TeX is called to process a complete LaTeX document, such as when you view(foo), where foo is
a complicated Sage object, too complicated for MathJax to handle. The command latex_extra_preamble isused to build the preamble of the complete LaTeX document, so the following illustrates how this is accomplished. As
usual note the need for the double-backslashes in the Python strings.
sage: latex.extra_macros( )sage: latex.extra_preamble( )sage: from sage.misc.latex import latex_extra_preamble
sage: print(latex_extra_preamble())\newcommand{\ZZ}{\Bold{Z}}...
\newcommand{\Bold}[1]{\mathbf{#1}}sage: latex.add_macro("\\newcommand{\\foo}{bar}")sage: print(latex_extra_preamble())\newcommand{\ZZ}{\Bold{Z}}...
\newcommand{\Bold}[1]{\mathbf{#1}}\newcommand{\foo}{bar}>>> from sage.all import *>>> latex.extra_macros( )>>> latex.extra_preamble( )>>> from sage.misc.latex import latex_extra_preamble>>> print(latex_extra_preamble())\newcommand{\ZZ}{\Bold{Z}}...
\newcommand{\Bold}[1]{\mathbf{#1}}>>> latex.add_macro("\\newcommand{\\foo}{bar}")>>> print(latex_extra_preamble())\newcommand{\ZZ}{\Bold{Z}}...
\newcommand{\Bold}[1]{\mathbf{#1}}\newcommand{\foo}{bar}Again, for larger or more complicated LaTeX expressions, it is possible to add packages (or anything else) to the preambleof the LaTeX file. Anything may be incorporated into the preamble with the latex.add_to_preamble command,and the specialized command latex.add_package_to_preamble_if_available will first check if a certain
package is actually available before trying to add it to the preamble.

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Here we add the geometry package to the preamble and use it to set the size of the region on the page that TeX will use(effectively setting the margins). As usual, note the need for the double-backslashes in the Python strings.
sage: from sage.misc.latex import latex_extra_preamble
sage: latex.extra_macros( )sage: latex.extra_preamble( )sage: latex.add_to_preamble( \\usepackage{geometry} )sage: latex.add_to_preamble( \\geometry{letterpaper,total={8in,10in}} )sage: latex.extra_preamble()\\usepackage{geometry}\\geometry{letterpaper,total={8in,10in}}sage: print(latex_extra_preamble())\usepackage{geometry}\geometry{letterpaper,total={8in,10in}}\newcommand{\ZZ}{\Bold{Z}}...
\newcommand{\Bold}[1]{\mathbf{#1}}>>> from sage.all import *>>> from sage.misc.latex import latex_extra_preamble>>> latex.extra_macros( )>>> latex.extra_preamble( )>>> latex.add_to_preamble( \\usepackage{geometry} )>>> latex.add_to_preamble( \\geometry{letterpaper,total={8in,10in}} )>>> latex.extra_preamble()\\usepackage{geometry}\\geometry{letterpaper,total={8in,10in}}>>> print(latex_extra_preamble())\usepackage{geometry}\geometry{letterpaper,total={8in,10in}}\newcommand{\ZZ}{\Bold{Z}}...
\newcommand{\Bold}[1]{\mathbf{#1}}

A particular package may be added along with a check on its existence, as follows. As an example, we just illustrate anattempt to add to the preamble a package that presumably does not exist.
sage: latex.extra_preamble( )sage: latex.extra_preamble()sage: latex.add_to_preamble( \\usepackage{foo-bar-unchecked} )sage: latex.extra_preamble()\\usepackage{foo-bar-unchecked}
sage: latex.add_package_to_preamble_if_available( foo-bar-checked )sage: latex.extra_preamble()\\usepackage{foo-bar-unchecked}>>> from sage.all import *>>> latex.extra_preamble( )>>> latex.extra_preamble()>>> latex.add_to_preamble( \\usepackage{foo-bar-unchecked} )>>> latex.extra_preamble()\\usepackage{foo-bar-unchecked}>>> latex.add_package_to_preamble_if_available( foo-bar-checked )>>> latex.extra_preamble()\\usepackage{foo-bar-unchecked}

Which dialect of TeX is used, and therefore the nature of the output and associated viewer, can also be customized.
Note: Sage includes almost everything you need to build and use Sage, but a significant exception is TeX itself. So in the120

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following situations you need to have a full TeX system installed, along with some associated conversion utilities. Many
versions of Linux have packages based on TeXLive, for macOS there is MacTeX and for Windows there is MiKTeX.
The latex.engine() command can be used to control if the system-wide executables latex, pdflatex or xelatex are employed. When view() is called and the engine is set to latex, a dvi file is produced and Sage will use advi viewer (like xdvi) to display the result. In contrast, using view() when the engine is set to pdflatex will producea PDF as the result and Sage will call your system’s utility for displaying PDF files (acrobat, okular, evince, etc.).
For your exercises with these facilities, there are some pre-packaged examples. To use these, it is necessary toimport the sage.misc.latex.latex_examples object, which is an instance of the sage.misc.latex.
LatexExamples class, as illustrated below. This class currently has examples of commutative diagrams, combinatorial graphs, knot theory and pstricks, which respectively exercise the following packages: xy, tkz-graph, xypic, pstricks.
After the import, use tab-completion on latex_examples to see the pre-packaged examples. Calling each example
will give you back some explanation about what is required to make the example render properly. To actually see theexamples, it is necessary to use view(foo) (once the preamble, engine, etc are all set properly).
sage: from sage.misc.latex import latex_examples
sage: foo = latex_examples.diagram()
sage: foo
LaTeX example for testing display of a commutative diagram producedby xypic.
To use, try to view this object -- it will not work. Now try
latex.add_to_preamble("\\usepackage[matrix,arrow,curve,cmtip]{xy}") ,and try viewing again. You should get a picture (a part of the diagram arisingfrom a filtered chain complex).
>>> from sage.all import *>>> from sage.misc.latex import latex_examples>>> foo = latex_examples.diagram()>>> foo
LaTeX example for testing display of a commutative diagram producedby xypic.
<BLANKLINE>
To use, try to view this object -- it will not work. Now try
latex.add_to_preamble("\\usepackage[matrix,arrow,curve,cmtip]{xy}") ,and try viewing again. You should get a picture (a part of the diagram arisingfrom a filtered chain complex).

For an example of how complicated LaTeX expressions can be processed, let us see the example of combinatorial graphs,that use tkz-graph LaTeX package.
Note: tkz-graph LaTeX package is built on top of the tikz front-end to the pgf library. Rendering combinatorial
graphs requires the pgf library, and the files tkz-graph.sty and tkz-berge.sty. It is highly likely that they
are already part of your system-wide TeX installation. Even if not, it should be easy to find instructions to install them.
First, we ensure that the relevant packages are included by adding them to the preamble of the LaTeX document.
sage: latex.extra_preamble( \\usepackage{tikz}\n\\usepackage{tkz-graph}\n....:\\usepackage{tkz-berge}\n\\usetikzlibrary{arrows,shapes} )>>> from sage.all import *>>> latex.extra_preamble( \\usepackage{tikz}\n\\usepackage{tkz-graph}\n...
\\usepackage{tkz-berge}\n\\usetikzlibrary{arrows,shapes} )5.3. Customizing LaTeX Processing

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The images of graphs do not form properly when a dvi file is used as an intermediate format, so it is best to set the latex
engine to the pdflatex executable:
sage: latex.engine( pdflatex )>>> from sage.all import *>>> latex.engine( pdflatex )At this point a command like view(graphs.CompleteGraph(4)) should produce a PDF with an appropriate
image of the complete graph 𝐾4 .
Actually the preliminary steps could be omitted as the preamble is automatically set up properly for graphs and pdflatex
is the default LaTeX engine in Sage. Try the command again after restarting Sage.
Note that there is a variety of options to affect how a graph is rendered in LaTeX via tkz-graph, which is outside thescope of this section. See the section LaTeX options for graphs of the Reference Manual for instructions and details.

5.4 SageTeX
SageTeX is a program available to further integrate TeX and Sage. It is a collection of TeX macros that allow a LaTeX
document to include instructions to have Sage compute various objects and format objects using the latex(). See Using
SageTeX for more information.

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6.1 Loading and Attaching Sage files
Next we illustrate how to load programs written in a separate file into Sage. Create a file called example.sage withthe following content:
print("Hello World")
print(2^3)

You can read in and execute example.sage file using the load command.
sage: load("example.sage")
Hello World8>>> from sage.all import *>>> load("example.sage")
Hello World8

You can also attach a Sage file to a running session using the attach command:sage: attach("example.sage")
Hello World8>>> from sage.all import *>>> attach("example.sage")
Hello World8

Now if you change example.sage and enter one blank line into Sage (i.e., hit return), then the contents ofexample.sage will be automatically reloaded into Sage.
In particular, attach automatically reloads a file whenever it changes, which is handy when debugging code, whereasload only loads a file once.
When Sage loads example.sage it converts it to Python, which is then executed by the Python interpreter. This conversion is minimal; it mainly involves wrapping integer literals in Integer() floating point literals in RealNumber(),
replacing ^’s by **’s, and replacing e.g., R.2 by R.gen(2). The converted version of example.sage is containedin the same directory as example.sage and is called example.sage.py. This file contains the following code:

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print("Hello World")
print(Integer(2)**Integer(3))

Integer literals are wrapped and the ^ is replaced by a **. (In Python ^ means “exclusive or” and ** means “exponentiation”.)(This preparsing is implemented in sage/misc/interpreter.py.)
You can paste multi-line indented code into Sage as long as there are newlines to make new blocks (this is not necessary
in files). However, the best way to enter such code into Sage is to save it to a file and use attach, as described above.

6.2 Creating Compiled Code
Speed is crucial in mathematical computations. Though Python is a convenient very high-level language, certain calculations can be several orders of magnitude faster than in Python if they are implemented using static types in a compiled
language. Some aspects of Sage would have been too slow if it had been written entirely in Python. To deal with this,Sage supports a compiled “version” of Python called Cython ([Cyt] and [Pyr]). Cython is simultaneously similar to both
Python and C. Most Python constructions, including list comprehensions, conditional expressions, code like += are allowed; you can also import code that you have written in other Python modules. Moreover, you can declare arbitrary C
variables, and arbitrary C library calls can be made directly. The resulting code is converted to C and compiled using a
C compiler.
In order to make your own compiled Sage code, give the file an .spyx extension (instead of .sage). If you areworking with the command-line interface, you can attach and load compiled code exactly like with interpreted code (atthe moment, attaching and loading Cython code is not supported with the notebook interface). The actual compilation isdone “behind the scenes” without your having to do anything explicit. The compiled shared object library is stored under$HOME/.sage/temp/hostname/pid/spyx. These files are deleted when you exit Sage.
NO Sage preparsing is applied to spyx files, e.g., 1/3 will result in 0 in a spyx file instead of the rational number 1/3. If
foo is a function in the Sage library, to use it from a spyx file import sage.all and use sage.all.foo.
import sage.alldef foo(n):return sage.all.factorial(n)6.2.1 Accessing C Functions in Separate Files
It is also easy to access C functions defined in separate *.c files. Here’s an example. Create files test.c and test.
spyx in the same directory with contents:
The pure C code: test.c
int add_one(int n) {return n + 1;}

The Cython code: test.spyx:
cdef extern from "test.c":
int add_one(int n)
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Then the following works:
sage: attach("test.spyx")
Compiling (...)/test.spyx...
sage: test(10)11>>> from sage.all import *>>> attach("test.spyx")
Compiling (...)/test.spyx...
>>> test(Integer(10))
11

If an additional library foo is needed to compile the C code generated from a Cython file, add the line clib foo to
the Cython source. Similarly, an additional C file bar can be included in the compilation with the declaration cfilebar.

6.3 Standalone Python/Sage Scripts
The following standalone Sage script factors integers, polynomials, etc:#!/usr/bin/env sage
import sys
if len(sys.argv) != 2:
print("Usage: %s <n>" % sys.argv[0])print("Outputs the prime factorization of n.")sys.exit(1)print(factor(sage_eval(sys.argv[1])))

In order to use this script, your SAGE_ROOT must be in your PATH. If the above script is called factor, here is an
example usage:$ ./factor 2006
2 * 17 * 59

6.4 Data Types
Every object in Sage has a well-defined type. Python has a wide range of basic built-in types, and the Sage library adds
many more. Some built-in Python types include strings, lists, tuples, ints and floats, as illustrated:sage: s = "sage"; type(s)<... str >sage: s = sage ; type(s)<... str >sage: s = [1,2,3,4]; type(s)<... list >sage: s = (1,2,3,4); type(s)<... tuple >sage: s = int(2006); type(s)# you can use either single or double quotes(continues on next page)

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(continued from previous page)<... int >sage: s = float(2006); type(s)<... float >>>> from sage.all import *>>> s = "sage"; type(s)<... str >>>> s = sage ; type(s)# you can use either single or double quotes<... str >>>> s = [Integer(1),Integer(2),Integer(3),Integer(4)]; type(s)<... list >>>> s = (Integer(1),Integer(2),Integer(3),Integer(4)); type(s)<... tuple >>>> s = int(Integer(2006)); type(s)<... int >>>> s = float(Integer(2006)); type(s)<... float >To this, Sage adds many other types. E.g., vector spaces:sage: V = VectorSpace(QQ, 1000000); V
Vector space of dimension 1000000 over Rational Field
sage: type(V)<class sage.modules.free_module.FreeModule_ambient_field_with_category >>>> from sage.all import *>>> V = VectorSpace(QQ, Integer(1000000)); V
Vector space of dimension 1000000 over Rational Field>>> type(V)<class sage.modules.free_module.FreeModule_ambient_field_with_category >Only certain functions can be called on V. In other math software systems, these would be called using the “functional”
notation foo(V,...). In Sage, certain functions are attached to the type (or class) of V, and are called using an
object-oriented syntax like in Java or C++, e.g., V.foo(...). This helps keep the global namespace from being polluted with tens of thousands of functions, and means that many different functions with different behavior can be named
foo, without having to use type-checking of arguments (or case statements) to decide which to call. Also, if you reuse
the name of a function, that function is still available (e.g., if you call something zeta, then want to compute the valueof the Riemann-Zeta function at 0.5, you can still type s=.5; s.zeta()).
sage: zeta = -1
sage: s=.5; s.zeta()-1.46035450880959>>> from sage.all import *>>> zeta = -Integer(1)>>> s=RealNumber( .5 ); s.zeta()-1.46035450880959

In some very common cases, the usual functional notation is also supported for convenience and because mathematicalexpressions might look confusing using object-oriented notation. Here are some examples.
sage: n = 2; n.sqrt()
sqrt(2)sage: sqrt(2)sqrt(2)(continues on next page)

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(continued from previous page)

sage: V = VectorSpace(QQ,2)
sage: V.basis()[(1, 0),(0, 1)]
sage: basis(V)[(1, 0),(0, 1)]
sage: M = MatrixSpace(GF(7), 2); M
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 7
sage: A = M([1,2,3,4]); A
[1 2]
[3 4]
sage: A.charpoly( x )x^2 + 2*x + 5
sage: charpoly(A, x )x^2 + 2*x + 5>>> from sage.all import *>>> n = Integer(2); n.sqrt()
sqrt(2)>>> sqrt(Integer(2))
sqrt(2)>>> V = VectorSpace(QQ,Integer(2))>>> V.basis()[(1, 0),(0, 1)]>>> basis(V)[(1, 0),(0, 1)]>>> M = MatrixSpace(GF(Integer(7)), Integer(2)); M
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 7>>> A = M([Integer(1),Integer(2),Integer(3),Integer(4)]); A
[1 2]
[3 4]
>>> A.charpoly( x )x^2 + 2*x + 5>>> charpoly(A, x )x^2 + 2*x + 5

To list all member functions for 𝐴, use tab completion. Just type A., then type the [tab] key on your keyboard, as
explained in Reverse Search and Tab Completion.

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6.5 Lists, Tuples, and Sequences
The list data type stores elements of arbitrary type. Like in C, C++, etc. (but unlike most standard computer algebra
systems), the elements of the list are indexed starting from 0:
sage: v = [2, 3, 5, x , SymmetricGroup(3)]; v[2, 3, 5, x , Symmetric group of order 3! as a permutation group]sage: type(v)<... list >sage: v[0]2sage: v[2]5>>> from sage.all import *>>> v = [Integer(2), Integer(3), Integer(5), x , SymmetricGroup(Integer(3))]; v[2, 3, 5, x , Symmetric group of order 3! as a permutation group]>>> type(v)<... list >>>> v[Integer(0)]2>>> v[Integer(2)]5(When indexing into a list, it is OK if the index is not a Python int!) A Sage Integer (or Rational, or anything with an__index__ method) will work just fine.
sage: v = [1,2,3]
sage: v[2]3sage: n = 2# Sage Integer
sage: v[n]# Perfectly OK!
3
sage: v[int(n)] # Also OK.
3>>> from sage.all import *>>> v = [Integer(1),Integer(2),Integer(3)]>>> v[Integer(2)]3>>> n = Integer(2)# Sage Integer>>> v[n]# Perfectly OK!
3>>> v[int(n)] # Also OK.
3

The range function creates a list of Python int’s (not Sage Integers):
sage: list(range(1, 15))
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]>>> from sage.all import *>>> list(range(Integer(1), Integer(15)))[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]This is useful when using list comprehensions to construct lists:

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sage: L = [factor(n) for n in range(1, 15)]
sage: L[1, 2, 3, 2^2, 5, 2 * 3, 7, 2^3, 3^2, 2 * 5, 11, 2^2 * 3, 13, 2 * 7]sage: L[12]13sage: type(L[12])<class sage.structure.factorization_integer.IntegerFactorization >sage: [factor(n) for n in range(1, 15) if is_odd(n)][1, 3, 5, 7, 3^2, 11, 13]>>> from sage.all import *>>> L = [factor(n) for n in range(Integer(1), Integer(15))]>>> L[1, 2, 3, 2^2, 5, 2 * 3, 7, 2^3, 3^2, 2 * 5, 11, 2^2 * 3, 13, 2 * 7]>>> L[Integer(12)]13>>> type(L[Integer(12)])<class sage.structure.factorization_integer.IntegerFactorization >>>> [factor(n) for n in range(Integer(1), Integer(15)) if is_odd(n)][1, 3, 5, 7, 3^2, 11, 13]

For more about how to create lists using list comprehensions, see [PyT].
List slicing is a wonderful feature. If L is a list, then L[m:n] returns the sublist of L obtained by starting at the 𝑚𝑡ℎelement and stopping at the (𝑛 − 1)𝑠𝑡 element, as illustrated below.
sage: L = [factor(n) for n in range(1, 20)]
sage: L[4:9][5, 2 * 3, 7, 2^3, 3^2]
sage: L[:4][1, 2, 3, 2^2]
sage: L[14:4][]
sage: L[14:][3 * 5, 2^4, 17, 2 * 3^2, 19]>>> from sage.all import *>>> L = [factor(n) for n in range(Integer(1), Integer(20))]>>> L[Integer(4):Integer(9)][5, 2 * 3, 7, 2^3, 3^2]>>> L[:Integer(4)][1, 2, 3, 2^2]>>> L[Integer(14):Integer(4)][]>>> L[Integer(14):][3 * 5, 2^4, 17, 2 * 3^2, 19]

Tuples are similar to lists, except they are immutable, meaning once they are created they can’t be changed.
sage: v = (1,2,3,4); v(1, 2, 3, 4)
sage: type(v)<... tuple >sage: v[1] = 5
Traceback (most recent call last):...
TypeError: tuple object does not support item assignment

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Tutorial, Release 10.4>>> from sage.all import *>>> v = (Integer(1),Integer(2),Integer(3),Integer(4)); v(1, 2, 3, 4)>>> type(v)<... tuple >>>> v[Integer(1)] = Integer(5)Traceback (most recent call last):...
TypeError: tuple object does not support item assignment

Sequences are a third list-oriented Sage type. Unlike lists and tuples, Sequence is not a built-in Python type. By default,
a sequence is mutable, but using the Sequence class method set_immutable, it can be set to be immutable, as thefollowing example illustrates. All elements of a sequence have a common parent, called the sequences universe.
sage: v = Sequence([1,2,3,4/5])
sage: v
[1, 2, 3, 4/5]
sage: type(v)<class sage.structure.sequence.Sequence_generic >sage: type(v[1])<class sage.rings.rational.Rational >sage: v.universe()
Rational Field
sage: v.is_immutable()False
sage: v.set_immutable()sage: v[0] = 3
Traceback (most recent call last):...
ValueError: object is immutable; please change a copy instead.
>>> from sage.all import *>>> v = Sequence([Integer(1),Integer(2),Integer(3),Integer(4)/Integer(5)])>>> v[1, 2, 3, 4/5]>>> type(v)<class sage.structure.sequence.Sequence_generic >>>> type(v[Integer(1)])<class sage.rings.rational.Rational >>>> v.universe()
Rational Field>>> v.is_immutable()
False>>> v.set_immutable()>>> v[Integer(0)] = Integer(3)Traceback (most recent call last):...
ValueError: object is immutable; please change a copy instead.

Sequences derive from lists and can be used anywhere a list can be used:
sage: v = Sequence([1,2,3,4/5])
sage: isinstance(v, list)
True
sage: list(v)[1, 2, 3, 4/5]
sage: type(list(v))(continues on next page)

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(continued from previous page)<...

list >>>> from sage.all import *>>> v = Sequence([Integer(1),Integer(2),Integer(3),Integer(4)/Integer(5)])>>> isinstance(v, list)
True>>> list(v)[1, 2, 3, 4/5]>>> type(list(v))<... list >As another example, basis for vector spaces are immutable sequences, since it’s important that you don’t change them.
sage: V = QQ^3; B = V.basis(); B[(1, 0, 0),(0, 1, 0),(0, 0, 1)]
sage: type(B)<class sage.structure.sequence.Sequence_generic >sage: B[0] = B[1]Traceback (most recent call last):...
ValueError: object is immutable; please change a copy instead.
sage: B.universe()
Vector space of dimension 3 over Rational Field>>> from sage.all import *>>> V = QQ**Integer(3); B = V.basis(); B[(1, 0, 0),(0, 1, 0),(0, 0, 1)]>>> type(B)<class sage.structure.sequence.Sequence_generic >>>> B[Integer(0)] = B[Integer(1)]Traceback (most recent call last):...
ValueError: object is immutable; please change a copy instead.
>>> B.universe()
Vector space of dimension 3 over Rational Field

6.6 Dictionaries
A dictionary (also sometimes called an associative array) is a mapping from ‘hashable’ objects (e.g., strings, numbers,
and tuples of such; see the Python documentation http://docs.python.org/tut/node7.html and http://docs.python.org/lib/
typesmapping.html for details) to arbitrary objects.
sage: d = {1:5,
sage: type(d)<... dict >sage :17, ZZ:GF(7)}(continues on next page)

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sage: list(d.keys())[1, sage , Integer Ring]sage: d[ sage ]17
sage: d[ZZ]
Finite Field of size 7
sage: d[1]5>>> from sage.all import *>>> d = {Integer(1):Integer(5),>>> type(d)<... dict >>>> list(d.keys())[1, sage , Integer Ring]>>> d[ sage ]17>>> d[ZZ]
Finite Field of size 7>>> d[Integer(1)]5sage :Integer(17), ZZ:GF(Integer(7))}

The third key illustrates that the indexes of a dictionary can be complicated, e.g., the ring of integers.
You can turn the above dictionary into a list with the same data:
sage: list(d.items())[(1, 5), ( sage , 17), (Integer Ring, Finite Field of size 7)]>>> from sage.all import *>>> list(d.items())[(1, 5), ( sage , 17), (Integer Ring, Finite Field of size 7)]

A common idiom is to iterate through the pairs in a dictionary:
sage: d = {2:4, 3:9, 4:16}
sage: [a*b for a, b in d.items()][8, 27, 64]>>> from sage.all import *>>> d = {Integer(2):Integer(4), Integer(3):Integer(9), Integer(4):Integer(16)}>>> [a*b for a, b in d.items()][8, 27, 64]

A dictionary is unordered, as the last output illustrates.

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6.7 Sets
Python has a built-in set type. The main feature it offers is very fast lookup of whether an element is in the set or not,along with standard set-theoretic operations.
sage: X = set([1,19, a ]);Y = set([1,1,1, 2/3])
sage: X# random sort order{1, 19, a }sage: X == set([ a , 1, 1, 19])True
sage: Y
{2/3, 1}
sage: a in X
True
sage: a in Y
False
sage: X.intersection(Y){1}>>> from sage.all import *>>> X = set([Integer(1),Integer(19), a ]);Y = set([Integer(1),Integer(1),˓→Integer(1), Integer(2)/Integer(3)])>>> X# random sort order{1, 19, a }>>> X == set([ a , Integer(1), Integer(1), Integer(19)])
True
>>> Y
{2/3, 1}>>> a in X
True>>> a in Y
False>>> X.intersection(Y){1}

Sage also has its own set type that is (in some cases) implemented using the built-in Python set type, but has a little bit ofextra Sage-related functionality. Create a Sage set using Set(...). For example,
sage: X = Set([1,19, a ]);Y = Set([1,1,1, 2/3])
sage: X# random sort order{ a , 1, 19}
sage: X == Set([ a , 1, 1, 19])True
sage: Y{1, 2/3}
sage: X.intersection(Y){1}
sage: print(latex(Y))\left\{1, \frac{2}{3}\right\}
sage: Set(ZZ)
Set of elements of Integer Ring>>> from sage.all import *>>> X = Set([Integer(1),Integer(19), a ]);˓→Integer(1), Integer(2)/Integer(3)])>>> X# random sort order

Y = Set([Integer(1),Integer(1),(continues on next page)

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(continued from previous page){ a , 1, 19}>>> X == Set([ a , Integer(1), Integer(1), Integer(19)])
True
>>> Y
{1, 2/3}>>> X.intersection(Y){1}>>> print(latex(Y))\left\{1, \frac{2}{3}\right\}>>> Set(ZZ)
Set of elements of Integer Ring

6.8 Iterators
Iterators are a recent addition to Python that are particularly useful in mathematics applications. Here are several examples;see [PyT] for more details. We make an iterator over the squares of the nonnegative integers up to 10000000.
sage: v = (n^2 for n in range(10000000))
sage: next(v)0sage: next(v)1sage: next(v)4>>> from sage.all import *>>> v = (n**Integer(2) for n in range(Integer(10000000)))>>> next(v)0>>> next(v)1>>> next(v)4

We create an iterate over the primes of the form 4𝑝 + 1 with 𝑝 also prime, and look at the first few values.
sage: w = (4*p + 1 for p in Primes() if is_prime(4*p+1))
sage: w
# in the next line, 0xb0853d6c is a random 0x number<generator object at 0xb0853d6c>
sage: next(w)13
sage: next(w)29
sage: next(w)53>>> from sage.all import *>>> w = (Integer(4)*p + Integer(1) for p in Primes() if is_
˓→prime(Integer(4)*p+Integer(1)))>>> w# in the next line, 0xb0853d6c is a random 0x number<generator object at 0xb0853d6c>>>> next(w)13(continues on next page)

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(continued from previous page)>>> next(w)29>>> next(w)53

Certain rings, e.g., finite fields and the integers have iterators associated to them:sage: [x for x in GF(7)][0, 1, 2, 3, 4, 5, 6]
sage: W = ((x,y) for x in ZZ for y in ZZ)
sage: next(W)(0, 0)
sage: next(W)(0, 1)
sage: next(W)(0, -1)>>> from sage.all import *>>> [x for x in GF(Integer(7))][0, 1, 2, 3, 4, 5, 6]>>> W = ((x,y) for x in ZZ for y in ZZ)>>> next(W)(0, 0)>>> next(W)(0, 1)>>> next(W)(0, -1)

6.9 Loops, Functions, Control Statements, and Comparisons
We have seen a few examples already of some common uses of for loops. In Python, a for loop has an indented
structure, such as>>> for i in range(5):...
print(i)
...
0
1
2
3
4

Note the colon at the end of the for statement (there is no “do” or “od” as in GAP or Maple), and the indentation before
the “body” of the loop, namely print(i). This indentation is important. In Sage, the indentation is automatically put
in for you when you hit enter after a “:”, as illustrated below.
sage: for i in range(5):....:print(i) # now hit enter twice
....:
0
1
2
3
4

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Tutorial, Release 10.4>>> from sage.all import *>>> for i in range(Integer(5)):...
print(i) # now hit enter twice
....:
0
1
2
3
4

The symbol = is used for assignment. The symbol == is used to check for equality:
sage: for i in range(15):....:
if gcd(i,15) == 1:....:
print(i)
....:
1
2
4
7
8
11
13
14
>>> from sage.all import *>>> for i in range(Integer(15)):...
if gcd(i,Integer(15)) == Integer(1):...
print(i)
....:
1
2
4
7
8
11
13
14

Keep in mind how indentation determines the block structure for if, for, and while statements:
sage: def legendre(a,p):....:is_sqr_modp=-1....:
for i in range(p):....:if a % p == i^2 % p:....:
is_sqr_modp=1....:
return is_sqr_modp
sage: legendre(2,7)
1sage: legendre(3,7)-1>>> from sage.all import *>>> def legendre(a,p):...
is_sqr_modp=-Integer(1)...
for i in range(p):(continues on next page)

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(continued from previous page)

...
...
...

if a % p == i**Integer(2) % p:
is_sqr_modp=Integer(1)return is_sqr_modp>>> legendre(Integer(2),Integer(7))
1>>> legendre(Integer(3),Integer(7))-1

Of course this is not an efficient implementation of the Legendre symbol! It is meant to illustrate various aspects of
Python/Sage programming. The function {kronecker}, which comes with Sage, computes the Legendre symbol efficiently
via a C-library call to PARI.
Finally, we note that comparisons, such as ==, !=, <=, >=, >, <, between numbers will automatically convert both
numbers into the same type if possible:
sage: 2 < 3.1; 3.1 <= 1
True
False
sage: 2/3 < 3/2;
3/2 < 3/1
True
True>>> from sage.all import *>>> Integer(2) < RealNumber( 3.1 ); RealNumber( 3.1 ) <= Integer(1)
True
False
>>> Integer(2)/Integer(3) < Integer(3)/Integer(2);Integer(3)/Integer(2) <␣
˓→Integer(3)/Integer(1)
True
True

Use bool for symbolic inequalities:
sage: x < x + 1
x < x + 1
sage: bool(x < x + 1)
True>>> from sage.all import *>>> x < x + Integer(1)x < x + 1>>> bool(x < x + Integer(1))
True

When comparing objects of different types in Sage, in most cases Sage tries to find a canonical coercion of both objects
to a common parent (see Parents, Conversion and Coercion for more details). If successful, the comparison is performed
between the coerced objects; if not successful, the objects are considered not equal. For testing whether two variables
reference the same object use is. As we see in this example, the Python int 1 is unique, but the Sage Integer 1 is not:sage: 1 is 2/2
False
sage: 1 is 1
False
sage: 1 == 2/2
True

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Tutorial, Release 10.4>>> from sage.all import *>>> Integer(1) is Integer(2)/Integer(2)False>>> Integer(1) is Integer(1)False>>> Integer(1) == Integer(2)/Integer(2)True

In the following two lines, the first equality is False because there is no canonical morphism → F5 , hence no canonicalway to compare the 1 in F5 to the 1 ∈ . In contrast, there is a canonical map → F5 , hence the second comparison isTrue. Note also that the order doesn’t matter.
sage: GF(5)(1) == QQ(1); QQ(1) == GF(5)(1)
False
False
sage: GF(5)(1) == ZZ(1); ZZ(1) == GF(5)(1)
True
True
sage: ZZ(1) == QQ(1)True>>> from sage.all import *>>> GF(Integer(5))(Integer(1)) == QQ(Integer(1)); QQ(Integer(1)) ==␣
˓→GF(Integer(5))(Integer(1))
False
False
>>> GF(Integer(5))(Integer(1)) == ZZ(Integer(1)); ZZ(Integer(1)) ==␣
˓→GF(Integer(5))(Integer(1))
True
True
>>> ZZ(Integer(1)) == QQ(Integer(1))
True

WARNING: Comparison in Sage is more restrictive than in Magma, which declares the 1 ∈ F5 equal to 1 ∈sage: magma( GF(5)!1 eq Rationals()!1 )true>>> from sage.all import *>>> magma( GF(5)!1 eq Rationals()!1 )true.

# optional - magma# optional - magma

6.10 Profiling“Premature optimization is the root of all evil.” - Donald Knuth
Section author: Martin Albrecht <malb@informatik.uni-bremen.de>
Sometimes it is useful to check for bottlenecks in code to understand which parts take the most computational time; thiscan give a good idea of which parts to optimize. Python and therefore Sage offers several profiling–as this process iscalled–options.
The simplest to use is the prun command in the interactive shell. It returns a summary describing which functions took
how much computational time. To profile (the currently slow! - as of version 1.0) matrix multiplication over finite fields,
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sage: k,a = GF(2**8, a ).objgen()sage: A = Matrix(k,10,10,[k.random_element() for _ in range(10*10)])>>> from sage.all import *>>> k,a = GF(Integer(2)**Integer(8), a ).objgen()>>> A = Matrix(k,Integer(10),Integer(10),[k.random_element() for _ in␣˓→range(Integer(10)*Integer(10))])sage: %prun B = A*A
32893 function calls in 1.100 CPU seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
12127 0.160
0.000
0.160 0.000 :0(isinstance)
2000 0.150
0.000
0.280 0.000 matrix.py:2235(__getitem__)
1000 0.120
0.000
0.370 0.000 finite_field_element.py:392(__mul__)1903 0.120
0.000
0.200 0.000 finite_field_element.py:47(__init__)
1900 0.090
0.000
0.220 0.000 finite_field_element.py:376(__compat)
900 0.080
0.000
0.260 0.000 finite_field_element.py:380(__add__)
1 0.0700.070
1.100 1.100 matrix.py:864(__mul__)2105 0.070
0.000
0.070 0.000 matrix.py:282(ncols)...
>>> from sage.all import *>>> %prun B = A*A
32893 function calls in 1.100 CPU seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
12127 0.160
0.000
0.160 0.000 :0(isinstance)
2000 0.150
0.000
0.280 0.000 matrix.py:2235(__getitem__)
1000 0.120
0.000
0.370 0.000 finite_field_element.py:392(__mul__)1903 0.120
0.000
0.200 0.000 finite_field_element.py:47(__init__)
1900 0.090
0.000
0.220 0.000 finite_field_element.py:376(__compat)
900 0.080
0.000
0.260 0.000 finite_field_element.py:380(__add__)
1 0.0700.070
1.100 1.100 matrix.py:864(__mul__)2105 0.070
0.000
0.070 0.000 matrix.py:282(ncols)...

Here ncalls is the number of calls, tottime is the total time spent in the given function (and excluding time made incalls to sub-functions), percall is the quotient of tottime divided by ncalls. cumtime is the total time spent inthis and all sub-functions (i.e., from invocation until exit), percall is the quotient of cumtime divided by primitive
calls, and filename:lineno(function) provides the respective data of each function. The rule of thumb here
is: The higher the function in that listing, the more expensive it is. Thus it is more interesting for optimization.
As usual, prun? provides details on how to use the profiler and understand the output.
The profiling data may be written to an object as well to allow closer examination:
sage: %prun -r A*A
sage: stats = _sage: stats?

6.10. Profiling

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Tutorial, Release 10.4>>> from sage.all import *>>> %prun -r A*A>>> stats = _>>> stats?

Note: entering stats = prun -r A\*A displays a syntax error message because prun is an IPython shell command,
not a regular function.
For a nice graphical representation of profiling data, you can use the hotshot profiler, a small script called hotshot2cachetree and the program kcachegrind (Unix only). The same example with the hotshot profiler:
sage: k,a = GF(2**8, a ).objgen()sage: A = Matrix(k,10,10,[k.random_element() for _ in range(10*10)])sage: import hotshot
sage: filename = "pythongrind.prof"
sage: prof = hotshot.Profile(filename, lineevents=1)>>> from sage.all import *>>> k,a = GF(Integer(2)**Integer(8), a ).objgen()>>> A = Matrix(k,Integer(10),Integer(10),[k.random_element() for _ in␣˓→range(Integer(10)*Integer(10))])>>> import hotshot>>> filename = "pythongrind.prof">>> prof = hotshot.Profile(filename, lineevents=Integer(1))
sage: prof.run("A*A")<hotshot.Profile instance at 0x414c11ec>
sage: prof.close()>>> from sage.all import *>>> prof.run("A*A")<hotshot.Profile instance at 0x414c11ec>>>> prof.close()This results in a file pythongrind.prof in the current working directory. It can now be converted to the cachegrind
format for visualization.
On a system shell, type
$ hotshot2calltree -o cachegrind.out.42 pythongrind.prof

The output file cachegrind.out.42 can now be examined with kcachegrind. Please note that the naming
convention cachegrind.out.XX needs to be obeyed.

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CHAPTERSEVENUSING SAGETEXThe SageTeX package allows you to embed the results of Sage computations into a LaTeX document. To use it, you willneed to “install” it first (see Make SageTeX known to TeX).

7.1 An example
Here is a very brief example of using SageTeX. The full documentation can be found in SAGE_ROOT/venv/share/doc/sagetex, where SAGE_ROOT is the directory where your Sage installation is located. That directory contains
the documentation and an example file. See SAGE_ROOT/venv/share/texmf/tex/latex/sagetex for somepossibly useful Python scripts.
To see how SageTeX works, follow the directions for installing SageTeX (in Make SageTeX known to TeX) and copy thefollowing text into a file named, say, st_example.tex:
Warning: The text below will have several errors about unknown control sequences if you are viewing this in the“live” help. Use the static version to see the correct text.

\documentclass{article}\usepackage{sagetex}\begin{document}
Using Sage\TeX, one can use Sage to compute things and put them into
your \LaTeX{} document. For example, there are$\sage{number_of_partitions(1269)}$ integer partitions of $1269$.
You don t need to compute the number yourself, or even cut and paste
it from somewhere.
Here s some Sage code:\begin{sageblock}f(x) = exp(x) * sin(2*x)\end{sageblock}
The second derivative of $f$ is\[\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}} \sage{f(x)} =\sage{diff(f, x, 2)(x)}.
\](continues on next page)141

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(continued from previous page)

Here s a plot of $f$ from $-1$ to $1$:
\sageplot{plot(f, -1, 1)}\end{document}

Run LaTeX on st_example.tex as usual. Note that LaTeX will have some complaints, which will include:
Package sagetex Warning: Graphics filesage-plots-for-st_example.tex/plot-0.eps on page 1 does not exist. Plotcommand is on input line 25.
Package sagetex Warning: There were undefined Sage formulas and/or
plots. Run Sage on st_example.sagetex.sage, and then run LaTeX on
st_example.tex again.

Notice that, in addition to the usual collection of files produced by LaTeX, there is a file called st_example.
sagetex.sage. That is a Sage script produced when you run LaTeX on st_example.tex. The warning message
told you to run Sage on st_example.sagetex.sage, so take its advice and do that. It will tell you to run LaTeX on
st_example.tex again, but before you do that, notice that a new file has been created: st_example.sagetex.
sout. That file contains the results of Sage’s computations, in a format that LaTeX can use to insert into your text. A
new directory containing an EPS file of your plot has also been created. Run LaTeX again and you’ll see that everythingthat Sage computed and plotted is now included in your document.
The different macros used above should be pretty easy to understand. A sageblock environment typesets your code
verbatim and also executes the code when you run Sage. When you do \sage{foo}, the result put into your document
is whatever you get from running latex(foo) inside Sage. Plot commands are a bit more complicated, but in theirsimplest form, \sageplot{foo} inserts the image you get from doing foo.save( filename.eps ).
In general, the mantra is:• run LaTeX on your .tex file;• run Sage on the generated .sage file;• run LaTeX again.
You can omit running Sage if you haven’t changed around any Sage commands in your document.
There’s a lot more to SageTeX, and since both Sage and LaTeX are complex, powerful tools, it’s a good idea to read the
documentation for SageTeX, which is in SAGE_ROOT/venv/share/doc/sagetex.

7.2 Make SageTeX known to TeX
Sage is largely self-contained, but some parts do need some intervention to work properly. SageTeX is one such part.
The SageTeX package allows one to embed computations and plots from Sage into a LaTeX document. SageTeX is
installed in Sage by default, but to use SageTeX with your LaTeX documents, you need to make your TeX installation
aware of it before it will work.
The key to this is that TeX needs to be able to find sagetex.sty, which can be found in SAGE_ROOT/venv/
share/texmf/tex/latex/sagetex/, where SAGE_ROOT is the directory where you built or installed Sage. If
TeX can find sagetex.sty, then SageTeX will work. There are several ways to accomplish this.
• The first and simplest way is simply to copy sagetex.sty into the same directory as your LaTeX document.
Since the current directory is always searched when typesetting a document, this will always work.

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There are a couple small problems with this, however: the first is that you will end up with many unnecessary copies
of sagetex.sty scattered around your computer. The second and more serious problem is that if you upgrade
Sage and get a new version of SageTeX, the Python code and LaTeX code for SageTeX may no longer match,causing errors.
• The second way is to use the TEXMFLOCAL environment variable. If you are using the bash shell, you can do$ export TEXMFLOCAL=SAGE_ROOT/venv/share/texmf$ mktexlsr# update kpathsea ls-R databases

where SAGE_ROOT is the location of your Sage installation. Thereafter, TeX and friends will find the SageTeX
style file. If you want to make this change persistent, you can add the 1st of the above lines to your .bashrc file.
If you are using a different shell, you may have to modify the above command to make the environment variableknown; see your shell’s documentation for how to do that.
If you ever move your Sage installation, or install a new version into a new directory, you’ll need to update theabove command to reflect the new value of SAGE_ROOT.
• The third (and best) way to make TeX aware of sagetex.sty is to copy that file into a convenient place in yourhome directory. In most TeX distributions, the texmf directory in your home directory is automatically searchedfor packages. To find out exactly what this directory is, do the following on the command line:$ kpsewhich -var-value=TEXMFHOME

which will print out a directory, such as /home/drake/texmf or /Users/drake/Library/texmf.
Copy the tex/ directory from SAGE_ROOT/venv/share/texmf/ into your home texmf directory with acommand like$ cp -R SAGE_ROOT/venv/share/texmf/tex TEXMFHOME

where SAGE_ROOT is, as usual, replaced with the location of your Sage installation and TEXMFHOME is the resultof the kpsewhich command above.
If you upgrade Sage and discover that SageTeX no longer works, you can simply repeat these steps and the Sage
and TeX parts of SageTeX will again be synchronized.
• For installation on a multiuser system, you just modify the above instructions appropriately to copy sagetex.
sty into a systemwide TeX directory. Instead of the directory TEXMFHOME, probably the best choice is to use
the result of$ kpsewhich -var-value=TEXMFLOCAL

which will likely produce something like /usr/local/share/texmf. Copy the tex directory as above into
the TEXMFLOCAL directory. Now you need to update TeX’s database of packages, which you can do simply byrunning$ texhash TEXMFLOCAL

as root, replacing TEXMFLOCAL appropriately. Now all users of your system will have access to the LaTeX
package, and if they can also run Sage, they will be able to use SageTeX.
Warning: it’s very important that the file sagetex.sty that LaTeX uses when typesetting your document matchthe version of SageTeX that Sage is using. If you upgrade your Sage installation, you really should delete all the oldversions of sagetex.sty floating around.
Because of this problem, we recommend copying the SageTeX files into your home directory’s texmf directory (thethird method above). Then there is only one thing you need to do (copy a directory) when you upgrade Sage to insure

7.2. Make SageTeX known to TeX

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that SageTeX will work properly.

7.3 SageTeX documentation
While not strictly part of installation, it bears mentioning here that the documentation for SageTeX is maintained in
SAGE_ROOT/venv/share/doc/sagetex/sagetex.pdf. There is also an example file in the same directory– see example.tex and example.pdf, the pre-built result of typesetting that file with LaTeX and Sage. You canalso get those files from the SageTeX page.

7.4 SageTeX and TeXLive
One potentially confusing issue is that the popular TeX distribution TeXLive includes SageTeX. This may seem nice, butwith SageTeX, it’s important that the Sage bits and LaTeX bits be synchronized – which is a problem in this case, since
TeXLive, as shipped by your OS distro, or package manager, might be out of sync with TeXLive distribution, and thelatter might also be out of sync with the current SageTeX.
Because of this, it is strongly recommended that you always install the LaTeX part of SageTeX from Sage, as describedabove. The instructions above will insure that both halves of SageTeX are compatible and will work properly.

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CHAPTER

EIGHTAFTERWORD

8.1 Why Python?
8.1.1 Advantages of Python
The primary implementation language of Sage is Python (see [Py]), though code that must be fast is implemented in acompiled language. Python has several advantages:• Object saving is well-supported in Python. There is extensive support in Python for saving (nearly) arbitrary
objects to disk files or a database.
• Excellent support for documentation of functions and packages in the source code, including automatic extractionof documentation and automatic testing of all examples. The examples are automatically tested regularly andguaranteed to work as indicated.
• Memory management: Python now has a well thought out and robust memory manager and garbage collector that
correctly deals with circular references, and allows for local variables in files.
• Python has many packages available now that might be of great interest to users of Sage: numerical analysis andlinear algebra, 2D and 3D visualization, networking (for distributed computations and servers, e.g., via twisted),
database support, etc.
• Portability: Python is easy to compile from source on most platforms in minutes.
• Exception handling: Python has a sophisticated and well thought out system of exception handling, whereby
programs gracefully recover even if errors occur in code they call.
• Debugger: Python includes a debugger, so when code fails for some reason, the user can access an extensive stack
trace, inspect the state of all relevant variables, and move up and down the stack.
• Profiler: There is a Python profiler, which runs code and creates a report detailing how many times and for how
long each function was called.
• A Language: Instead of writing a new language for mathematics as was done for Magma, Maple, Mathematica,
Matlab, GP/PARI, GAP, Macaulay 2, Simath, etc., we use the Python language, which is a popular computer
language that is being actively developed and optimized by hundreds of skilled software engineers. Python is amajor open-source success story with a mature development process (see [PyDev]).

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8.1.2 The Pre-Parser: Differences between Sage and Python
Some mathematical aspects of Python can be confusing, so Sage behaves differently from Python in several ways.
• Notation for exponentiation: ** versus ^. In Python, ^ means “xor”, not exponentiation, so in Python we have>>> 2^8
10>>> 3^2
1>>> 3**29This use of ^ may appear odd, and it is inefficient for pure math research, since the “exclusive or” function is rarely
used. For convenience, Sage pre-parses all command lines before passing them to Python, replacing instances of^ that are not in strings with **:sage: 2^8256
sage: 3^2
9sage: "3^2"
3^2>>> from sage.all import *>>> Integer(2)**Integer(8)256>>> Integer(3)**Integer(2)9>>> "3^2"
3^2

The bitwise xor operator in Sage is ^^. This also works for the inplace operator ^^=:
sage: 3^^2
1sage: a = 2
sage: a ^^= 8
sage: a10>>> from sage.all import *>>> Integer(3)^Integer(2)1>>> a = Integer(2)>>> a ^= Integer(8)
>>> a
10

• Integer division: The Python expression 2/3 does not behave the way mathematicians might expect. In Python2,
if m and n are ints, then m/n is also an int, namely the quotient of m divided by n. Therefore 2/3=0. In Python3,
2/3 returns the floating point number 0.6666.... In both Python2 and Python3, // is the Euclidean divisionand 2//3 returns 0.
We deal with this in the Sage interpreter, by wrapping integer literals in Integer( ) and making division aconstructor for rational numbers. For example:

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sage: 2/3
2/3
sage: (2/3).parent()
Rational Field
sage: 2//30
sage: int(2)/int(3) # not tested, python20>>> from sage.all import *>>> Integer(2)/Integer(3)2/3>>> (Integer(2)/Integer(3)).parent()
Rational Field>>> Integer(2)//Integer(3)0>>> int(Integer(2))/int(Integer(3)) # not tested, python20• Long integers: Python has native support for arbitrary precision integers, in addition to C-int’s. These are significantly slower than what GMP provides, and have the property that they print with an L at the end to distinguishthem from int’s (and this won’t change any time soon). Sage implements arbitrary precision integers using the GMP
C-library, and these print without an L.
Rather than modifying the Python interpreter (as some people have done for internal projects), we use the Python language
exactly as is, and write a pre-parser for IPython so that the command line behavior of IPython is what a mathematician
expects. This means any existing Python code can be used in Sage. However, one must still obey the standard Python
rules when writing packages that will be imported into Sage.
(To install a Python library, for example that you have found on the Internet, follow the directions, but run sage
-python instead of python. Very often this means typing sage -python setup.py install.)

8.2 I would like to contribute somehow. How can I?
If you would like to contribute to Sage, your help will be greatly appreciated! It can range from substantial code contributions to adding to the Sage documentation to reporting bugs.
Browse the Sage web page for information for developers; among other things, you can find a long list of Sage-related
projects ordered by priority and category. The Sage Developer’s Guide has helpful information, as well, and you can also
check out the sage-devel Google group.

8.3 How do I reference Sage?
If you write a paper using Sage, please reference computations done with Sage by including[Sage] SageMath, the Sage Mathematics Software System (Version 8.7),
The Sage Developers, 2019, https://www.sagemath.org.

in your bibliography (replacing 8.7 with the version of Sage you used). Moreover, please attempt to track down what
components of Sage are used for your computation, e.g., PARI?, GAP?, Singular? Maxima? and also cite those systems.
If you are in doubt about what software your computation uses, feel free to ask on the sage-devel Google group. See
Univariate Polynomials for further discussion of this point.

8.2. I would like to contribute somehow. How can I?

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If you happen to have just read straight through this tutorial, and have some sense of how long it took you, please let us
know on the sage-devel Google group.
Have fun with Sage!

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CHAPTERNINE

APPENDIX

9.1 Arithmetical binary operator precedence
What is 3^2*4 + 2%5? The value (38) is determined by this “operator precedence table”. The table below is basedon the table in § 5.14 of the Python Language Reference Manual by G. Rossum and F. Drake. the operations are listed
here in increasing order of precedence.
Operators

Description

or
and
not
in, not in
is, is not>, <=, >, >=, ==, !=+, *, /, %**, ^boolean or
boolean andboolean notmembership
identity testcomparison
addition, subtraction
multiplication, division, remainder
exponentiation

Therefore, to compute 3^2*4 + 2%5, Sage brackets the computation this way: ((3^2)*4) + (2%5). Thus, firstcompute 3^2, which is 9, then compute both (3^2)*4 and 2%5, and finally add these.

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CHAPTERTENBIBLIOGRAPHY

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Chapter 10. Bibliography

CHAPTERELEVEN

INDICES AND TABLES• genindex• search

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Chapter 11. Indices and tables

BIBLIOGRAPHY[Cyt]

Cython, http://www.cython.org[GAP]

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4; 2005, https://www.
gap-system.org[GAPkg]GAP Packages, https://www.gap-system.org/Packages/packages.html[GP]

PARI/GP, https://pari.math.u-bordeaux.fr/[Mag]

Magma, http://magma.maths.usyd.edu.au/magma/[Max]

Maxima, http://maxima.sf.net/[NagleEtAl2004] Nagle, Saff, and Snider. Fundamentals of Differential Equations. 6th edition, Addison-Wesley, 2004.
[Py]

The Python language, http://www.python.org/[PyB]

The Python Beginner’s Guide, https://wiki.python.org/moin/BeginnersGuide[PyDev]

Python Developer’s Guide, https://docs.python.org/devguide/[PyLR]

Python Library Reference, https://docs.python.org/3/library/index.html[Pyr]

Pyrex, http://www.cosc.canterbury.ac.nz/ greg/python/Pyrex/[PyT]

The Python Tutorial, https://docs.python.org/3/tutorial/[SA]

Sage web site, https://www.sagemath.org/[Si]

G.-M. Greuel, G. Pfister, and H. Schönemann. Singular 3.0. A Computer Algebra System for Polynomial
Computations. Center for Computer Algebra, University of Kaiserslautern (2005). https://www.singular.
uni-kl.de[SJ]

William Stein, David Joyner, Sage: System for Algebra and Geometry Experimentation, Comm. Computer
Algebra {39} (2005) 61-64.

[ThreeJS]

three.js, http://threejs.org

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Bibliography

INDEXEEDITOR, 95
environment variable
EDITOR, 95

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