# 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?

edit retag close merge delete

Sort by » oldest newest most voted

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)

more

Then this object T can be used, for example to map points from one curve to the other,

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

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

more

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.