ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 30 Sep 2016 04:35:10 -0500Elliptic curve from Weierstrass form to minimal formhttps://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/ 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?Sun, 25 Sep 2016 20:52:05 -0500https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/Answer by John Cremona for <p>I used PARI code <code>ellglobalreduce</code> and <code>ellchangecurve</code> to change my Elliptic curve : <code>y^2=x^3-3267x+45630</code> to the minimal form. Can I do that with SAGE?</p>
https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?answer=34967#post-id-34967Sage 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)
Tue, 27 Sep 2016 02:21:38 -0500https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?answer=34967#post-id-34967Comment by Sha for <p>Sage can give you the isomorphism too, in a separate operation:</p>
<pre><code>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)
</code></pre>
https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?comment=34996#post-id-34996@john Thank you so much. It is great to know SAGE can do this as well.Fri, 30 Sep 2016 04:35:10 -0500https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?comment=34996#post-id-34996Comment by John Cremona for <p>Sage can give you the isomorphism too, in a separate operation:</p>
<pre><code>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)
</code></pre>
https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?comment=34968#post-id-34968Then this object T can be used, for example to map points from one curve to the other,Tue, 27 Sep 2016 02:22:23 -0500https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?comment=34968#post-id-34968Answer by B r u n o for <p>I used PARI code <code>ellglobalreduce</code> and <code>ellchangecurve</code> to change my Elliptic curve : <code>y^2=x^3-3267x+45630</code> to the minimal form. Can I do that with SAGE?</p>
https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?answer=34944#post-id-34944Does `E.minimal_model()` compute what you need (where `E` is your curve)?Mon, 26 Sep 2016 02:42:58 -0500https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?answer=34944#post-id-34944Comment by Sha for <p>Does <code>E.minimal_model()</code> compute what you need (where <code>E</code> is your curve)?</p>
https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?comment=34962#post-id-34962Yes 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.Mon, 26 Sep 2016 19:22:21 -0500https://ask.sagemath.org/question/34941/elliptic-curve-from-weierstrass-form-to-minimal-form/?comment=34962#post-id-34962