Weierstrass equation of an elliptic curve

Weierstrass equation of an elliptic curve

Recall that an

elliptic curve

over a field

K

is a projective

nonsingular

curve

E

defined over

K

of genus

1

together with a point

O

∈

E

defined over

K

.

Definition

.

Let

K

be an arbitrary field. A

Weierstrass equation

for an elliptic curve

E

/

K

is an

equation

of the form:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

where

a

1

,

a

2

,

a

3

,

a

4

,

a

6

are

constants

in

K

.

All elliptic curves have a Weierstrass model in

ℙ

2

⁢

(

K

)

, the

projective plane

over

K

. This is a simple application of the

http://planetmath.org/node/RiemannRochTheorem

Riemann Roch

theorem

for curves:

Theorem

.

Let

E

be an elliptic curve defined over a field

K

. Then there exists rational functions

x

,

y

∈

K

⁢

(

E

)

such that the map

ψ

:

E

→

P

2

⁢

(

K

)

sending

P

to

[

x

⁢

(

P

)

,

y

⁢

(

P

)

,

1

]

is an

isomorphism

of

E

/

K

to the

projective curve

given by

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

where

a

1

,

a

2

,

a

3

,

a

4

,

a

6

are constants in

K

.

Moreover, the following

proposition

specifies any possible change of variables.

Proposition 1

.

Let

E

/

K

be an elliptic curve given by a Weierstrass model of the form:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

with

a

i

∈

K

. Then:

1.

The only change of variables

(

x

,

y

)

↦

(

x

′

,

y

′

)

preserving the projective point

[

0

,

1

,

0

]

and which also result in a Weierstrass equation, are of the form:

x

=

u

2

⁢

x

′

+

r

,

y

=

u

3

⁢

y

′

+

s

⁢

u

2

⁢

x

′

+

t

with

u

,

r

,

s

,

t

∈

K

and

u

≠

0

.

2.

Any two Weierstrass equations for

E

/

K

differ by a change of variables of the form given in

(

1

)

.

Once we have one Weierstrass model for a given elliptic curve

E

/

K

, and as long as the

characteristic

of

K

is not

2

or

3

, there exists a change of variables (of the form given in the previous proposition) which simplifies the model considerably.

Corollary

.

Let

K

be a field of characteristic different from

2

or

3

. Let

E

be an elliptic curve defined over

K

. Then there exists a Weierstrass model for

E

of the form:

y

2

=

x

3

+

A

⁢

x

+

B

where

A

,

B

are elements of

K

.

Finally, remember that the

j

-

invariant

of an elliptic curve is invariant under isomorphism, but the

discriminant

depends on the model chosen.

Proposition 2

.

Let

E

/

K

be an elliptic curve and let

E

1

:

y

2

+

a

1

⁢

x

⁢

y

+

a

3

⁢

y

=

x

3

+

a

2

⁢

x

2

+

a

4

⁢

x

+

a

6

,

E

2

:

y

′

⁣

2

+

a

1

⁢

x

′

⁢

y

′

+

a

3

⁢

y

′

=

x

′

⁣

3

+

a

2

⁢

x

′

⁣

2

+

a

4

⁢

x

′

+

a

6

be two distinct Weierstrass models for

E

/

K

. Then (by Prop.

1

) there exists a change of variables

(

x

,

y

)

↦

(

x

′

,

y

′

)

of the form:

x

=

u

2

⁢

x

′

+

r

,

y

=

u

3

⁢

y

′

+

s

⁢

u

2

⁢

x

′

+

t

with

u

,

r

,

s

,

t

∈

K

and

u

≠

0

. Moreover,

j

⁢

(

E

1

)

=

j

⁢

(

E

2

)

, i.e. the

j

invariants are equal (

j

⁢

(

E

)

is defined in

http://planetmath.org/node/JInvariant

this entry) and

Δ

⁢

(

E

1

)

=

u

12

⁢

Δ

⁢

(

E

2

)

, where

Δ

⁢

(

E

i

)

is the discriminant (as defined in

http://planetmath.org/node/JInvariant

here).

Title

Weierstrass equation of an elliptic curve

Canonical name

WeierstrassEquationOfAnEllipticCurve

Date of creation

2013-03-22 15:48:00

Last modified on

2013-03-22 15:48:00

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 11G05

Classification

msc 14H52

Classification

msc 11G07

Synonym

Weierstrass model

Defines

Weierstrass equation

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