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How to show that the following points are independent on elliptic curve $y^2=x^3-82x$ using Sage or any other software.

There is some determinant of certain matrix which I need to compute constructed from Néron-Tate height pairing. If this determinant is non-zero, then the points are independent. How to write a code to find this determinant? Three points on above curve are:

$[(-8,12),(-1,9),(49/4,231/8)]$

mathisgood
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  • I'm afraid this question is not suitable for this site, although it's related to mathematics. – WhatsUp Dec 11 '19 at 05:57
  • I saw it in the papers of Elliptic curves. I am sure those work on elliptic curves use Sage and that makes it relevant to this website. – mathisgood Dec 11 '19 at 06:10
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    I think you can use this

    Try the code below here.

    E = EllipticCurve([-82,0]); pts = [E(-8,12),E(-1,9),E(49/4,231/8)]; M = E.height_pairing_matrix(points = pts); print(M); print(M.det());

     – Yong Hao Ng Dec 11 '19 at 06:12
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    In particular the determinant is large so they are independent. If it's very small but not zero it's taken to be likely dependent from my understanding (This is from "Elliptic Curves, Modular Forms and their L-functions", page 48). Probably the methods cannot guarantee computing a zero determinant? – Yong Hao Ng Dec 11 '19 at 06:23
  • @WhatsUp Questions about using CASes are expressly on-topic (except for those about Mathematica, which has a dedicated SE site). https://math.stackexchange.com/help/on-topic – Travis Willse Dec 11 '19 at 06:29
  • @TravisWillse I see, I thought it would be more suitable for sage's own forum. – WhatsUp Dec 11 '19 at 06:38
  • @WhatsUp SAGE's elliptic curve support is one of its (relative and absolute) strengths. I would be surprised if this question weren't already effectively answered there (or possibly even in the documentation). – Travis Willse Dec 11 '19 at 06:41
  • @YongHaoNg Thank you very much. It works. – mathisgood Dec 11 '19 at 06:46

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