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Elliptic curve from Weierstrass form to minimal form

asked 2016-09-25 20:52:05 -0500

Sha gravatar image

I used PARI code ellglobalreduce and ellchangecurve to change my Elliptic curve : y^2=x^3-3267x+45630 to the minimal form. Can I do that with SAGE?

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answered 2016-09-27 02:21:38 -0500

John Cremona gravatar image

Sage can give you the isomorphism too, in a separate operation:

sage: E=EllipticCurve([-3267,45630]); E
  Elliptic Curve defined by y^2 = x^3 - 3267*x + 45630 over Rational Field
sage: Emin = E.minimal_model(); Emin
  Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2*x over Rational Field
sage: T = E.isomorphism_to(Emin); T
  Generic morphism:
    From: Abelian group of points on Elliptic Curve defined by y^2 = x^3 - 3267*x + 45630 over Rational Field
    To:   Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2*x over Rational Field
    Via:  (u,r,s,t) = (6, 15, 3, 0)
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Then this object T can be used, for example to map points from one curve to the other,

John Cremona gravatar imageJohn Cremona ( 2016-09-27 02:22:23 -0500 )edit

@john Thank you so much. It is great to know SAGE can do this as well.

Sha gravatar imageSha ( 2016-09-30 04:35:10 -0500 )edit
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answered 2016-09-26 02:42:58 -0500

B r u n o gravatar image

updated 2016-09-26 02:43:16 -0500

Does E.minimal_model() compute what you need (where E is your curve)?

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Yes it does. Thank you so much! But in my opinion PARI does a better job as it gives the map as well in terms of [u,r,s,t]. Anyway thanks.

Sha gravatar imageSha ( 2016-09-26 19:22:21 -0500 )edit

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Asked: 2016-09-25 20:52:05 -0500

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Last updated: Sep 27 '16