ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 22 Oct 2019 04:42:14 -0500Elliptic curves - morphismhttp://ask.sagemath.org/question/48457/elliptic-curves-morphism/ Consider the example from the documentation:
sage: R.<u,v,t> = QQ[]
sage: Jacobian(u^3+v^3+t, variables=[u,v])
Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over
Multivariate Polynomial Ring in u, v, t over Rational Field
how to obtain the morphism in this case?Tue, 22 Oct 2019 03:04:13 -0500http://ask.sagemath.org/question/48457/elliptic-curves-morphism/Answer by rburing for <p>Consider the example from the documentation:</p>
<pre><code>sage: R.<u,v,t> = QQ[]
sage: Jacobian(u^3+v^3+t, variables=[u,v])
Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over
Multivariate Polynomial Ring in u, v, t over Rational Field
</code></pre>
<p>how to obtain the morphism in this case?</p>
http://ask.sagemath.org/question/48457/elliptic-curves-morphism/?answer=48458#post-id-48458You can do this:
sage: T.<t> = FunctionField(QQ)
sage: R.<u,v> = T[]
sage: f = u^3+v^3+t
sage: h = Jacobian(f, curve=Curve(f)); h
Scheme morphism:
From: Affine Plane Curve over Rational function field in t over Rational Field defined by u^3 + v^3 + t
To: Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over Rational function field in t over Rational Field
Defn: Defined on coordinates by sending (u, v) to
((-t^3)*u^4*v^4 + (-t^4)*u^4*v + (-t^4)*u*v^4 : 1/2*t^3*u^6*v^3 + (-1/2*t^3)*u^3*v^6 + (-1/2*t^4)*u^6 + 1/2*t^4*v^6 + 1/2*t^5*u^3 + (-1/2*t^5)*v^3 : t^3*u^3*v^3)
Tue, 22 Oct 2019 04:01:22 -0500http://ask.sagemath.org/question/48457/elliptic-curves-morphism/?answer=48458#post-id-48458Comment by castor for <p>You can do this:</p>
<pre><code>sage: T.<t> = FunctionField(QQ)
sage: R.<u,v> = T[]
sage: f = u^3+v^3+t
sage: h = Jacobian(f, curve=Curve(f)); h
Scheme morphism:
From: Affine Plane Curve over Rational function field in t over Rational Field defined by u^3 + v^3 + t
To: Elliptic Curve defined by y^2 = x^3 + (-27/4*t^2) over Rational function field in t over Rational Field
Defn: Defined on coordinates by sending (u, v) to
((-t^3)*u^4*v^4 + (-t^4)*u^4*v + (-t^4)*u*v^4 : 1/2*t^3*u^6*v^3 + (-1/2*t^3)*u^3*v^6 + (-1/2*t^4)*u^6 + 1/2*t^4*v^6 + 1/2*t^5*u^3 + (-1/2*t^5)*v^3 : t^3*u^3*v^3)
</code></pre>
http://ask.sagemath.org/question/48457/elliptic-curves-morphism/?comment=48461#post-id-48461Thank you very much, it work well.Tue, 22 Oct 2019 04:42:14 -0500http://ask.sagemath.org/question/48457/elliptic-curves-morphism/?comment=48461#post-id-48461