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                                                                     Exhibit 4.1

                            CENTERPOINT ENERGY, INC.

                                       To

                    JPMORGAN CHASE BANK, NATIONAL ASSOCIATION
                         (formerly JPMorgan Chase Bank),

                                     Trustee

                               ------------------

                          SUPPLEMENTAL INDENTURE NO. 5

                          Dated as of December 13, 2004

                                -----------------

                    2.875% Convertible Senior Notes due 2024

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                            CENTERPOINT ENERGY, INC.

                          SUPPLEMENTAL INDENTURE NO. 5

                    2.875% Convertible Senior Notes due 2024

      SUPPLEMENTAL INDENTURE No. 5, dated as of December 13, 2004, between
CENTERPOINT ENERGY, INC., a Texas corporation (the "Company"), and JPMORGAN
CHASE BANK, NATIONAL ASSOCIATION (formerly JPMorgan Chase Bank), as Trustee (the
"Trustee").

                                    RECITALS

      The Company has heretofore executed and delivered to the Trustee an
Indenture, dated as of May 19, 2003 (the "Original Indenture" and, as previously
and hereby supplemented and amended, the "Indenture"), providing for the
issuance from time to time of one or more series of the Company's Securities.

      The Company provided for the establishment of one new series of Securities
designated as the "2.875% Convertible Senior Notes due 2024" (the "Notes"), the
form and substance of such Notes and the terms, provisions and conditions
thereof set forth in Supplemental Indenture No. 4 dated as of December 17, 2003
between the Company and the Trustee ("Supplemental Indenture No. 4").
Capitalized terms used herein shall have the meaning specified herein or in
Supplemental Indenture No. 4, as the case may be.

      Section 802 of Supplemental Indenture No. 4 permits the Company, at its
option, to settle conversions of the Company's Notes through either a Share
Settlement, a Cash Settlement or a Combined Settlement. Pursuant to the terms of
the Indenture, the Company seeks to amend Supplemental Indenture No. 4 to
eliminate the Company's right to settle any conversion of the Notes through a
Share Settlement.

      Subparagraph (5) of Section 901 of the Original Indenture provides that
the Company and the Trustee may, without the consent of the Holders, enter into
an indenture supplemental to the Indenture to add to, change or eliminate any of
the provisions of the Indenture in respect of one or more series of Securities
if such addition, change or elimination does not adversely affect the interests
of any holder of Securities.

      For consideration, the adequacy and sufficiency of which is hereby
acknowledged by the parties hereto, it is mutually covenanted and agreed, for
the equal and proportionate benefit of the Holders of the Notes, as follows:

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                                   ARTICLE I

                              Relation to Indenture

      Section 101 Relation to Indenture. This Supplemental Indenture No. 5
amends Supplemental Indenture No. 4 as provided in ARTICLE II below and
constitutes an integral part of the Original Indenture.

                                   ARTICLE II

                    Amendment of Supplemental Indenture No. 4

      Section 201 Amendment of Section 801. Section 801 of Supplemental
Indenture No. 4 is hereby amended and restated to read as follows:

                  "A Holder may convert its Notes for Common Stock at any time
during which the conditions stated in paragraph 10 of the Notes are met. The
number of shares of Common Stock issuable upon conversion of a Note per $1,000
principal amount (the "Conversion Rate") shall be that set forth in paragraph 10
in the Notes, subject to adjustment as herein set forth. The initial Conversion
Rate is 78.0640 shares of Common Stock issuable upon conversion of $1,000
principal amount of Notes. Upon conversion of a Note pursuant to this Article
VIII, the Company shall satisfy its obligations to deliver cash or a combination
of cash and shares of Common Stock, together with any cash payment for
fractional shares, in accordance with this Article VIII (the "Conversion
Obligation").

      A Holder may convert a portion of the principal amount of Notes if the
portion is $1,000 or a multiple of $1,000."

      Section 202 Amendment of Section 802. Section 802 of Supplemental
Indenture No. 4 is hereby amended and restated to read as follows:

      "(a) To convert Notes, a Holder must satisfy the requirements in this
Section 802 and in paragraph 10 of the Notes. The date on which the Holder
satisfies all those requirements is the conversion date (the "Conversion Date").
As soon as practicable after the Conversion Date, and subject to Section 802(b),
the Company shall satisfy its Conversion Obligation upon conversion of a Note by
delivering to the Holder, at the Company's election, either cash or a
combination of cash and shares of Common Stock as follows:

            (1) If the Company elects to satisfy the entire Conversion
      Obligation in cash ("Cash Settlement"), then the Company will deliver to
      such Holder cash in an amount equal to the product of (i) (x) the
      aggregate principal amount of Notes to be converted by such Holder divided
      by 1,000 multiplied by (y) the Conversion Rate in effect on the Conversion
      Date, and (ii) the average of the Last Reported Sale Prices of Common
      Stock for the five Trading Day period commencing on the third Trading Day
      following the Conversion Date (the "Applicable Stock Price"); or

            (2) If the Company elects to satisfy a portion of the Conversion
      Obligation in cash (the "Partial Cash Amount") and a portion in shares of
      Common Stock (together

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      with the Partial Cash Amount, a "Combined Settlement"), then the Company
      will deliver to such Holder such Partial Cash Amount, plus a number of
      full shares of Common Stock equal to (i) the amount of the Cash Settlement
      determined pursuant to clause (1) above minus such Partial Cash Amount
      divided by (B) the Applicable Stock Price, together with cash in lieu of
      any fractional shares determined pursuant to Section 803 based on the
      Applicable Stock Price. In no event shall the Partial Cash Amount equal
      less than the aggregate principal amount of the Notes to be converted.

      (b) Upon receipt of the conversion notice from a Holder by the Company or
the Conversion Agent:

            (1) If such notice is received by the Conversion Agent, the
      Conversion Agent shall promptly notify the Company.

            (2) The Company will notify the Trustee in writing, who shall
      promptly notify such Holder, of the dollar amount to be satisfied in cash
      at any time on or prior to the date that is two Business Days following
      the Conversion Date (the "Settlement Notice Period").

            (3) Such Holder may retract the conversion notice at any time during
      the two Business Day period beginning on the Business Day after the
      Settlement Notice Period (the "Conversion Retraction Period") by written
      notice to the Trustee. If such Holder has not retracted the conversion
      notice during the Conversion Retraction Period, then Cash Settlement or
      Combined Settlement will occur, through the Conversion Agent, on or prior
      to the third Business Day following the determination of the Applicable
      Stock Price. If such Holder timely retracts its conversion notice, then
      such Holder shall be deemed to be a Holder of Notes as if a conversion
      notice had never been delivered by such Holder for such Notes.

      (c) Upon Combined Settlement, the Person in whose name the certificate is
registered shall be treated as a stockholder of record on and after the
Conversion Date; provided, however, that no surrender of Notes on any date when
the stock transfer books of the Company shall be closed shall be effective to
constitute the Person or Persons entitled to receive the shares of Common Stock
upon such conversion as the record holder or holders of such shares of Common
Stock on such date, but such surrender shall be effective to constitute the
Person or Persons entitled to receive such shares of Common Stock as the record
holder or holders thereof for all purposes at the close of business on the next
succeeding day on which such stock transfer books are open; and provided further
that such conversion shall be at the Conversion Rate in effect on the date that
such Notes shall have been surrendered for conversion, as if the stock transfer
books of the Company had not been closed. Upon conversion of Notes, such Person
shall no longer be a Holder of such Notes.

      (d) No payment or adjustment shall be made for dividends on or other
distributions with respect to any Common Stock except as provided in Section 806
or as otherwise provided in this Indenture.

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      (e) On conversion of Notes, that portion of the principal amount and
accrued and unpaid interest, including Contingent Interest, if any, with respect
to the converted portion of the Notes shall not be canceled, extinguished or
forfeited, but rather shall be deemed to be paid in full to the Holder thereof
through the delivery of cash or a combination of cash and shares of Common Stock
(together with the cash payment, if any, in lieu of fractional shares) in
exchange for the portion of the Notes being converted pursuant to the provisions
hereof. Notwithstanding the conversion of any Notes, the Holders of the Notes
and any Common Stock issuable upon conversion thereof will continue to be
entitled to receive Additional Amounts in accordance with the Registration
Rights Agreement.

      (f) If a Holder converts more than one Note at the same time, the number
of shares of Common Stock issuable upon the conversion shall be based on the
total principal amount of the Notes converted. Upon surrender of a Note that is
converted in part, the Company shall execute, and the Trustee or the
Authenticating Agent shall authenticate and deliver to the Holder, a new Note in
an authorized denomination equal in principal amount to the unconverted portion
of the Note surrendered. If the last day on which Notes may be converted is a
legal holiday in a place where a Conversion Agent is located, the Notes may be
surrendered to that Conversion Agent on the next succeeding day that is not a
legal holiday."

      Section 203 Addition of Section 817. A new Section 817 is hereby added to
Supplemental Indenture No. 4 and reads as follows:

      "Section 817 Notice to Holders of Change of Settlement Provisions. Within
30 days of making any change to any provision in this Article VIII, the Company
shall provide a notice describing such change to Holders of the Notes. The
Company shall file with the Trustee and the Conversion Agent such notice."

      Section 204 Notes Deemed Conformed. As of the date hereof, the provisions
of the Notes shall be deemed to be conformed, without the necessity for any
reissuance or exchange of such Note or any other action on the part of the
Holders of Notes, the Company or the Trustee, so as to reflect this Supplemental
Indenture No. 5.

                                  ARTICLE III

                            Miscellaneous Provisions

      Section 301 The Indenture, as supplemented and amended by this
Supplemental Indenture No. 5, is in all respects hereby adopted, ratified and
confirmed.

      Section 302 This Supplemental Indenture No. 5 may be executed in any
number of counterparts, each of which shall be an original, but such
counterparts shall together constitute but one and the same instrument.

      Section 303 THIS SUPPLEMENTAL INDENTURE NO. 5 AND EACH NOTE SHALL BE
DEEMED TO BE A CONTRACT MADE UNDER THE LAWS OF THE STATE OF NEW YORK AND SHALL
BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH THE LAWS OF THE STATE OF NEW
YORK WITHOUT REGARD TO CONFLICTS OF LAWS PRINCIPLES THEREOF.

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      Section 304 If any provision in this Supplemental Indenture No. 5 limits,
qualifies or conflicts with another provision hereof which is required to be
included herein by any provisions of the Trust Indenture Act, such required
provision shall control.

      Section 305 In case any provision in this Supplemental Indenture No. 5 or
the Notes shall be invalid, illegal or unenforceable, the validity, legality and
enforceability of the remaining provisions shall not in any way be affected or
impaired thereby.

      Section 306 The recitals contained herein shall be taken as the statements
of the Company, and the Trustee does not assume any responsibility for their
correctness. The Trustee makes no representations as to the validity or
sufficiency of this Supplemental Indenture No. 5 or of the Notes.

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      IN WITNESS WHEREOF, the parties hereto have caused this Supplemental
Indenture No. 5 to be duly executed, as of the day and year first written above.

                                       CENTERPOINT ENERGY, INC.

                                       By: /s/ Gary Whitlock
                                           -------------------------------------
                                           Gary Whitlock
                                           Executive Vice President and Chief
                                           Financial Officer and

Attest:

/s/ Richard B. Dauphin
----------------------------------
Richard B. Dauphin
Assistant Corporate Secretary

(SEAL)

                                       JPMORGAN CHASE BANK,
                                       NATIONAL ASSOCIATION, as Trustee

                                       By: /s/ Carol Logan
                                           -------------------------------------
                                           Carol Logan
                                           Vice President and Trust Officer

(SEAL)

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