Document:

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                                                                    EXHIBIT 10.4

                                                                  EXECUTION COPY

                      FIRST AMENDMENT TO WARRANT AGREEMENT

                  THIS FIRST AMENDMENT TO WARRANT AGREEMENT (this "Amendment")
dated as of February 17, 2000 is between Brigham Exploration Company, a Delaware
corporation ("Issuer") and Bank of Montreal, a Canadian bank (the "Warrant
Holder" or "BMO").

                                    RECITALS

                  A. Brigham Oil & Gas, L.P., a Delaware limited partnership
(the "Borrower"), BMO and Societe Generale Southwest Agency (the "Prior
Lenders") previously entered into that certain Credit Agreement dated as of
January 26, 1998, as amended by First Amendment to Credit Agreement dated as of
August 20, 1998, Second Amendment to Credit Agreement dated as of March 26,
1999, Third Amendment to Credit Agreement dated as of July 19, 1999 and Fourth
Amendment to Credit Agreement dated as of October 7, 1999 (such Credit Agreement
as amended called the "Prior Credit Agreement"), pursuant to which the Prior
Lenders agreed to make certain loans and extensions of credit to the Borrower as
evidenced by those certain promissory notes in the aggregate principal amount of
$75,000,000 (the "Prior Notes").

                  B. By Partial Assignment of Notes of even date herewith Shell
Capital Inc. ("SCI") has acquired an interest in the Prior Notes and a like
interest in all security for the payment of the Prior Notes and the performance
of the Borrower's obligations under the Prior Credit Agreement; and

                  C. The Prior Lenders and SCI (collectively, the "Lenders")
have agreed to provide senior secured debt in the amount of up to $75,000,000
consisting of a rearrangement of the Prior Notes and advances to the Borrower,
and the Borrower, BMO as Agent (the "Agent") and the Lenders are entering into
that certain Amended and Restated Credit Agreement (the "Credit Agreement") of
even date herewith which amends and restates in its entirety the Prior Credit
Agreement and Prior Notes; and

                  D. Issuer and BMO now desire to amend certain provisions of
the Warrant Agreement dated as of July 19, 1999 between Issuer and BMO (the
"Warrant Agreement") in recognition of the benefits Issuer will obtain under the
Credit Agreement.

                  NOW THEREFORE, in consideration of the mutual covenants and
agreements herein contained, Issuer and BMO hereby agree that the Warrant
Agreement shall be amended as follows:

                  Section 1. Certain Definitions. As used in this Amendment, the
terms "Agent", "Amendment", "Borrower", "Credit Agreement", "Issuer", "BMO",
"SCI", "Lenders", "Prior Credit Agreement", "Prior Lenders," "Prior Notes", and
"Warrant Agreement" shall have the meanings indicated above; and unless
otherwise defined herein, all terms beginning with a capital letter which are
defined in the Warrant Agreement shall have the same meanings herein as therein
unless the context hereof otherwise requires.

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                  Section 2. Amendments to Warrant Agreement.

                  Section 2.1 The definition of Current Warrant Price in section
1. of the Warrant Agreement is hereby deleted in its entirety and the following
is substituted therefor:

                  "Current Warrant Price" shall mean, in respect of a share of
                  Common Stock at any date herein specified, two dollars and two
                  cents ($2.02) per share of Common Stock, subject to adjustment
                  from time to time as provided in this Agreement."

                  Section 2.2 Exhibit A to the Warrant Agreement (Form of
Warrant Certificate) is amended as follows:

                  (a)      to change the reference "of two and 25/100 dollars
                           ($2.25)" to "of two and 02/100 dollars ($2.02)";

                  (b)      to add the following language after the date "July
                           19, 1999, to the next to the last paragraph of the
                           Form of Warrant Certificate: "as amended by that
                           certain First Amendment to Warrant Agreement dated as
                           of February 17, 2000".

                  Section 2.3 All references to the "Credit Agreement" in the
Warrant Agreement shall henceforth refer to the Credit Agreement as defined in
this Amendment, as such Credit Agreement is from time to time further amended,
supplemented or restated.

                  Section 2.4 The second sentence of Section 4.3(b) of the
Warrant Agreement is hereby deleted in its entirety and the following is
substituted therefor:

                  "No adjustment of the number of shares of Common Stock for
                  which a Warrant shall be exercised shall be made under Section
                  4.3(a) of this Agreement upon the issuance of any Additional
                  Shares of Common Stock which are issued pursuant to the
                  exercise of any warrants or other subscription or purchase
                  rights or pursuant to the exercise of any conversion or
                  exchange rights in any Convertible Securities (i) if any such
                  adjustment shall previously have been made upon the issuance
                  of such Convertible Securities (or upon the issuance of any
                  such warrants or other rights) pursuant to Section 4.4 or
                  Section 4.5 of this Agreement, (ii) if no adjustment was
                  required pursuant to such sections upon the issuance of such
                  Convertible Securities, warrants or other rights or (iii) in
                  the event the issuance of such Convertible Securities,
                  warrants or other rights predates or is of the same date as
                  this Agreement, if no adjustment would have been required
                  pursuant to such sections upon such issuance had this
                  Agreement been in effect."

                  Section 3. Representations and Warranties. Issuer hereby
reaffirms that as of the effective date of this Amendment, the representations
and warranties made by the Issuer in the Warrant Agreement will be true and
correct as though made on and as of the date of this Amendment.

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                  Section 4. Ratification. Issuer hereby expressly ratifies and
affirms its obligations under the Warrant Agreement as amended by this Amendment
and agrees that the Warrant Agreement as amended by this Amendment remains in
full force and effect.

                  Section 5. Issuance of Warrant Certificate. Contemporaneously
with the execution of this Amendment, Issuer will deliver to BMO a Warrant
Certificate reflecting the amendments described in Section 2.2 hereof.

                  Section 6. Governing Law. This Amendment and the rights and
obligations of the parties hereunder and under the Warrant Agreement shall be
construed in accordance with and be governed by the laws of the State of Texas
and the United States of America.

                  Section 7. Descriptive Headings, Etc. The descriptive headings
of the several Sections of this Amendment are inserted for convenience only and
shall not be deemed to affect the meaning or construction of any of the
provisions hereof.

                  Section 8. Counterparts. This Amendment may be executed in any
number of counterparts and by different parties on separate counterparts and all
of such counterparts shall together constitute one and the same instrument.

                           [SIGNATURES ON NEXT PAGE.]

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                  IN WITNESS WHEREOF, the parties hereto have caused this
Amendment to be duly executed and delivered and effective as of the date first
above written.

                                        BANK OF MONTREAL, as Warrant Holder

                                        By:
                                           -------------------------------------
                                        Name: Thomas E. McGraw
                                        Title: Director

                                        BRIGHAM EXPLORATION COMPANY,
                                        As Issuer

                                        By:
                                           -------------------------------------
                                        Name: Karen E. Lynch
                                        Title: Vice President

                                       S<PAGE>   1
                                                                    EXHIBIT 10.5

                                                                  EXECUTION COPY

                      FIRST AMENDMENT TO WARRANT AGREEMENT

         THIS FIRST AMENDMENT TO WARRANT AGREEMENT (this "Amendment") dated as
of February 17, 2000 is between Brigham Exploration Company, a Delaware
corporation ("Issuer") and Societe Generale, Southwest Agent, a bank organized
under the laws of the Republic of France acting through its Southwest Agency
(the "Warrant Holder" or "Soc-Gen").

                                    RECITALS

         A. Brigham Oil & Gas, L.P., a Delaware limited partnership (the
"Borrower"), Bank of Montreal ("BMO") and Soc-Gen (the "Prior Lenders")
previously entered into that certain Credit Agreement dated as of January 26,
1998, as amended by First Amendment to Credit Agreement dated as of August 20,
1998, Second Amendment to Credit Agreement dated as of March 26, 1999, Third
Amendment to Credit Agreement dated as of July 19, 1999 and Fourth Amendment to
Credit Agreement dated as of October 7, 1999 (such Credit Agreement as amended
called the "Prior Credit Agreement"), pursuant to which the Prior Lenders agreed
to make certain loans and extensions of credit to the Borrower as evidenced by
those certain promissory notes in the aggregate principal amount of $75,000,000
(the "Prior Notes").

         B. By Partial Assignment of Notes of even date herewith Shell Capital
Inc. ("SCI") has acquired an interest in the Prior Notes and a like interest in
all security for the payment of the Prior Notes and the performance of the
Borrower's obligations under the Prior Credit Agreement; and

         C. The Prior Lenders and SCI (collectively, the "Lenders") have agreed
to provide senior secured debt in the amount of up to $75,000,000 consisting of
a rearrangement of the Prior Notes and advances to the Borrower, and the
Borrower, BMO as Agent (the "Agent") and the Lenders are entering into that
certain Amended and Restated Credit Agreement (the "Credit Agreement") of even
date herewith which amends and restates in its entirety the Prior Credit
Agreement and Prior Notes; and

         D. Issuer and Soc-Gen now desire to amend certain provisions of the
Warrant Agreement dated as of July 19, 1999 between Issuer and Soc-Gen (the
"Warrant Agreement") in recognition of the benefits Issuer will obtain under the
Credit Agreement.

         NOW THEREFORE, in consideration of the mutual covenants and agreements
herein contained, Issuer and Soc-Gen hereby agree that the Warrant Agreement
shall be amended as follows:

         Section 1. Certain Definitions. As used in this Amendment, the terms
"Agent", "Amendment", "Borrower", "Credit Agreement", "Issuer", "BMO", "SCI",
"Lenders", "Prior Credit Agreement", "Prior Lenders," "Prior Notes", "Soc-Gen"
and "Warrant Agreement" shall have the meanings indicated above; and unless
otherwise defined herein, all terms

<PAGE>   2

beginning with a capital letter which are defined in the Warrant Agreement shall
have the same meanings herein as therein unless the context hereof otherwise
requires.

         Section 2. Amendments to Warrant Agreement.

                  Section 2.1 The definition of Current Warrant Price in section
1. of the Warrant Agreement is hereby deleted in its entirety and the following
is substituted therefor:

                  "Current Warrant Price" shall mean, in respect of a share of
                  Common Stock at any date herein specified, two dollars and two
                  cents ($2.02) per share of Common Stock, subject to adjustment
                  from time to time as provided in this Agreement."

                  Section 2.2 Exhibit A to the Warrant Agreement (Form of
Warrant Certificate) is amended as follows:

                  (a)      to change the reference "of two and 25/100 dollars
                           ($2.25)" to "of two and 02/100 dollars ($2.02)";

                  (b)      to add the following language after the date "July
                           19, 1999, to the next to the last paragraph of the
                           Form of Warrant Certificate: "as amended by that
                           certain First Amendment to Warrant Agreement dated as
                           of February 17, 2000".

                  Section 2.3 All references to the "Credit Agreement" in the
Warrant Agreement shall henceforth refer to the Credit Agreement as defined in
this Amendment, as such Credit Agreement is from time to time further amended,
supplemented or restated.

                  Section 2.4 The second sentence of Section 4.3(b) of the
Warrant Agreement is hereby deleted in its entirety and the following is
substituted therefor:

                  "No adjustment of the number of shares of Common Stock for
                  which a Warrant shall be exercised shall be made under Section
                  4.3(a) of this Agreement upon the issuance of any Additional
                  Shares of Common Stock which are issued pursuant to the
                  exercise of any warrants or other subscription or purchase
                  rights or pursuant to the exercise of any conversion or
                  exchange rights in any Convertible Securities (i) if any such
                  adjustment shall previously have been made upon the issuance
                  of such Convertible Securities (or upon the issuance of any
                  such warrants or other rights) pursuant to Section 4.4 or
                  Section 4.5 of this Agreement, (ii) if no adjustment was
                  required pursuant to such sections upon the issuance of such
                  Convertible Securities, warrants or other rights or (iii) in
                  the event the issuance of such Convertible Securities,
                  warrants or other rights predates or is of the same date as
                  this Agreement, if no adjustment would have been required
                  pursuant to such sections upon such issuance had this
                  Agreement been in effect."

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<PAGE>   3

         Section 3. Representations and Warranties. Issuer hereby reaffirms that
as of the effective date of this Amendment, the representations and warranties
made by the Issuer in the Warrant Agreement will be true and correct as though
made on and as of the date of this Amendment.

         Section 4. Ratification. Issuer hereby expressly ratifies and affirms
its obligations under the Warrant Agreement as amended by this Amendment and
agrees that the Warrant Agreement as amended by this Amendment remains in full
force and effect.

         Section 5. Issuance of Warrant Certificate. Contemporaneously with the
execution of this Amendment, Issuer will deliver to Soc-Gen a Warrant
Certificate reflecting the amendments described in Section 2.2 hereof.

         Section 6. Governing Law. This Amendment and the rights and obligations
of the parties hereunder and under the Warrant Agreement shall be construed in
accordance with and be governed by the laws of the State of Texas and the United
States of America.

         Section 7. Descriptive Headings, Etc. The descriptive headings of the
several Sections of this Amendment are inserted for convenience only and shall
not be deemed to affect the meaning or construction of any of the provisions
hereof.

         Section 8. Counterparts. This Amendment may be executed in any number
of counterparts and by different parties on separate counterparts and all of
such counterparts shall together constitute one and the same instrument.

                           [SIGNATURES ON NEXT PAGE.]

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         IN WITNESS WHEREOF, the parties hereto have caused this Amendment to be
duly executed and delivered and effective as of the date first above written.

                                              SOCIETE GENERALE SOUTHWEST AGENCY,
                                              as Warrant Holder

                                              By:
                                                     ---------------------------
                                              Name:  Mark A. Cox
                                              Title: Director

                                              BRIGHAM EXPLORATION COMPANY,
                                              As Issuer

                                              By:
                                                     ---------------------------
                                              Name:  Karen E. Lynch
                                              Title: Vice President

                                       S

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