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.Sun, 12 Dec 2021 04:27:06 +0100Integral points on a cubic curve which is not in Weierstrass formhttps://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/ How can I compute the integral points on a cubic curve which is not in Weierstrass form? For example something like x^3 + y^3 + z^3 = 6xyz. I have tried using EllipticCurve_from_cubic, but this automatically transforms it into an equivalent (over Q) curve in Weierstrass form, which doesn't preserve the integral points.Sat, 11 Dec 2021 09:43:10 +0100https://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/Comment by slelievre for <p>How can I compute the integral points on a cubic curve which is not in Weierstrass form? For example something like x^3 + y^3 + z^3 = 6xyz. I have tried using EllipticCurve_from_cubic, but this automatically transforms it into an equivalent (over Q) curve in Weierstrass form, which doesn't preserve the integral points.</p>
https://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/?comment=60209#post-id-60209Welcome to Ask Sage! Thank you for your question.Sat, 11 Dec 2021 14:25:07 +0100https://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/?comment=60209#post-id-60209Answer by benwh1 for <p>How can I compute the integral points on a cubic curve which is not in Weierstrass form? For example something like x^3 + y^3 + z^3 = 6xyz. I have tried using EllipticCurve_from_cubic, but this automatically transforms it into an equivalent (over Q) curve in Weierstrass form, which doesn't preserve the integral points.</p>
https://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/?answer=60213#post-id-60213Wait, never mind. I just remembered that this is a homogeneous curve in 3 variables rather than a projective curve in 2 variables, so the integral points of this curve are the same as the rational points of the projective curve. So it doesn't matter that EllipticCurve_from_cubic rewrites it in Weierstrass form.
A.<x,y,z> = PolynomialRing(QQ, 3)
phi = EllipticCurve_from_cubic(x^3+y^3+z^3-6*x*y*z)
C = phi.codomain()
P = C.gens()[0]
P2 = phi.inverse()(5*P)
P2.clear_denominators()Sun, 12 Dec 2021 01:43:58 +0100https://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/?answer=60213#post-id-60213Comment by slelievre for <p>Wait, never mind. I just remembered that this is a homogeneous curve in 3 variables rather than a projective curve in 2 variables, so the integral points of this curve are the same as the rational points of the projective curve. So it doesn't matter that EllipticCurve_from_cubic rewrites it in Weierstrass form.</p>
<pre><code>A.<x,y,z> = PolynomialRing(QQ, 3)
phi = EllipticCurve_from_cubic(x^3+y^3+z^3-6*x*y*z)
C = phi.codomain()
P = C.gens()[0]
P2 = phi.inverse()(5*P)
P2.clear_denominators()
</code></pre>
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to mark the question as answered.Sun, 12 Dec 2021 04:27:06 +0100https://ask.sagemath.org/question/60207/integral-points-on-a-cubic-curve-which-is-not-in-weierstrass-form/?comment=60214#post-id-60214