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How to convert output of 'isomorphism_to' to transformation rule

asked 2019-12-23 15:21:21 -0600

azerbajdzan gravatar image

updated 2020-05-28 11:36:20 -0600

FrédéricC gravatar image

Input:

C = EllipticCurve([0,0,0,-((250)/3),-(1249/27)])
print(C)
Cmin=C.minimal_model()
print(Cmin)
Cmin.isomorphism_to(C)

Output:

Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
Elliptic Curve defined by y^2 = x^3 + x^2 - 83*x - 74 over Rational Field
Generic morphism:
  From: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 - 83*x - 74 over Rational Field
  To:   Abelian group of points on Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
  Via:  (u,r,s,t) = (1, -1/3, 0, 0)

But I want explicit transformation like $(x,y) = (u^2 x+r , u^3 y + s u^2 x + t)$, in our case it would be $(x, y) = (-(1/3) + x, y)$ instead of just showing $(u,r,s,t) = (1, -1/3, 0, 0)$.

EDIT:

What is wrong with the page? My question looked OK in preview window, but when sent there was gibberish inside $$.

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answered 2019-12-26 09:41:15 -0600

azerbajdzan gravatar image

updated 2020-01-02 04:47:53 -0600

Input:

from sage.schemes.elliptic_curves.weierstrass_morphism import *
C = EllipticCurve([0,0,0,-((250)/3),-(1249/27)])
print(C)
Cmin=C.minimal_model()
print(Cmin)
(u,r,s,t)=isomorphisms(Cmin,C,True)
var('x y')
[u^2*x+r,u^3*y+s*u^2*x+t]

Output:

Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
Elliptic Curve defined by y^2 = x^3 + x^2 - 83*x - 74 over Rational Field
[x - 1/3, y]
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Asked: 2019-12-23 15:21:21 -0600

Seen: 77 times

Last updated: Jan 02