ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 26 Mar 2020 22:10:47 +0100What is the point 𝑃1=(−1,3) on the y^2=7𝑥4+𝑥3+𝑥2+𝑥+3 curve and the point corresponding to P on the x^2=x^3−250/3x−124927 curve?https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/What is the point 𝑃1=(−1,3) on the y^2=7𝑥^4+𝑥^3+𝑥^2+𝑥+3 curve and the point corresponding to P on the x^2=x^3−250/3x−124927 curve?
(𝐸1:𝑦2=7𝑥4+𝑥3+𝑥2+𝑥+3,𝑃1=(−1,3) can be transformed to 𝐸2:𝑣2=𝑢3−250𝑢3−124927)
Thanks
Thu, 26 Mar 2020 16:08:06 +0100https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/Answer by nbruin for <p>What is the point 𝑃1=(−1,3) on the y^2=7𝑥^4+𝑥^3+𝑥^2+𝑥+3 curve and the point corresponding to P on the x^2=x^3−250/3x−124927 curve?
(𝐸1:𝑦2=7𝑥4+𝑥3+𝑥2+𝑥+3,𝑃1=(−1,3) can be transformed to 𝐸2:𝑣2=𝑢3−250𝑢3−124927)
Thanks</p>
https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?answer=50383#post-id-50383The question is, unfortunately, not well-defined. The curves mentioned, while isomorphic, are not uniquely so. First you should specify which point on E1 is supposed to be mapped to the "origin" (0:1:0) on "E2", and then there is still a choice of sign.
The routine `Jacobian` mentioned in another answer does not give you a (birational) isomorphism; it only expresses "E1" as a cover of "E2". There is even more choice in that.
As an example:
sage: P2.<x,y,z>=ProjectiveSpace(QQ,2)
sage: C=Curve(3*x^3+4*y^3+5*z^3)
sage: phi=Jacobian(C,morphism=True)
sage: phi.codomain()
Elliptic Curve defined by y^2 = x^3 - 24300 over Rational Field
This is Selmer's famous example: the curve C isn't isomorphic to the elliptic curve over Q, because C does not have rational points. The map phi computed is of degree larger than 1 (presumable degree 9, but I didn't check)Thu, 26 Mar 2020 22:10:47 +0100https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?answer=50383#post-id-50383Answer by rburing for <p>What is the point 𝑃1=(−1,3) on the y^2=7𝑥^4+𝑥^3+𝑥^2+𝑥+3 curve and the point corresponding to P on the x^2=x^3−250/3x−124927 curve?
(𝐸1:𝑦2=7𝑥4+𝑥3+𝑥2+𝑥+3,𝑃1=(−1,3) can be transformed to 𝐸2:𝑣2=𝑢3−250𝑢3−124927)
Thanks</p>
https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?answer=50374#post-id-50374See [Construct elliptic curves as Jacobians](http://doc.sagemath.org/html/en/reference/curves/sage/schemes/elliptic_curves/jacobian.html) in the manual.
sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)
Edit: in this way you certainly get a map, but not necessarily an isomorphism; see the other answer.Thu, 26 Mar 2020 16:30:24 +0100https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?answer=50374#post-id-50374Comment by nbruin for <p>See <a href="http://doc.sagemath.org/html/en/reference/curves/sage/schemes/elliptic_curves/jacobian.html">Construct elliptic curves as Jacobians</a> in the manual.</p>
<pre><code>sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)
</code></pre>
<p>Edit: in this way you certainly get a map, but not necessarily an isomorphism; see the other answer.</p>
https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?comment=50375#post-id-50375Note, though, that the map phi is not uniquely determined. You can compose phi with a translation on E1 and/or negation.Thu, 26 Mar 2020 17:40:23 +0100https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?comment=50375#post-id-50375Comment by Gamzeee for <p>See <a href="http://doc.sagemath.org/html/en/reference/curves/sage/schemes/elliptic_curves/jacobian.html">Construct elliptic curves as Jacobians</a> in the manual.</p>
<pre><code>sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)
</code></pre>
<p>Edit: in this way you certainly get a map, but not necessarily an isomorphism; see the other answer.</p>
https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?comment=50380#post-id-50380sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1= Curve(y^2z^2+15x^4-1516z^4)
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 + 14400*x over Rational Field
sage: phi(E1(2,0,1))
(0 : 1 : 0)
But the (0 : 1 : 0) point is not on the E2 curve. What should I do?Thu, 26 Mar 2020 18:32:21 +0100https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?comment=50380#post-id-50380Comment by rburing for <p>See <a href="http://doc.sagemath.org/html/en/reference/curves/sage/schemes/elliptic_curves/jacobian.html">Construct elliptic curves as Jacobians</a> in the manual.</p>
<pre><code>sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E1 = Curve(y^2*z^2 - (7*𝑥^4+𝑥^3*z+𝑥^2*z^2+𝑥*z^3+3*z^4))
sage: phi = Jacobian(E1, morphism=True)
sage: E2 = phi.codomain(); E2
Elliptic Curve defined by y^2 = x^3 - 250/3*x - 1249/27 over Rational Field
sage: phi(E1(-1,3,1))
(-71/9 : 296/27 : 1)
</code></pre>
<p>Edit: in this way you certainly get a map, but not necessarily an isomorphism; see the other answer.</p>
https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?comment=50381#post-id-50381@Gamzeee yes it is on the curve; it is the point at infinity. Maybe @nbruin can suggest how to get a different `phi`.Thu, 26 Mar 2020 18:33:30 +0100https://ask.sagemath.org/question/50373/what-is-the-point-p1-13-on-the-y27x4x3x2x3-curve-and-the-point-corresponding-to-p-on-the-x2x3-2503x-124927-curve/?comment=50381#post-id-50381