I am new to branch of elliptic curve and algebraic number theory .I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3-6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052}) $. Some suggestions such as algorithm or softwares will be helpful.

  • 4
    $\begingroup$ This question appears to be off-topic because it is about using the software Sage. Please ask in a Sage forum on question/answer site. $\endgroup$ Aug 29, 2014 at 16:11
  • $\begingroup$ This may be a hard to compute example. Sage indeed gets stuck. Magma reports a lower bound of 1 for the rank, so you might be able to coax it to give you a point of infinite order. $\endgroup$ Aug 29, 2014 at 17:35
  • $\begingroup$ Try ask.sagemath.org/questions $\endgroup$
    – GH from MO
    Aug 29, 2014 at 17:42
  • 6
    $\begingroup$ I think one prefers to use the algebraic method to compute the 2-Selmer group, as opposed to the invariant method. In Sage, this might be implemented by Simon in PARI. The 2-Selmer group has order 4. It is reasonably easy to find one point on a cover, mapping to an $x$-coordinate $6321363053/3418801$. It is a bit harder to find a second independent point, but one has $x$-coordinate $77367184671463906202575142600893705817/4363223875270025353581185386334464$. $\endgroup$ Aug 30, 2014 at 1:51
  • 1
    $\begingroup$ This is similar to a question Kevin Acres had: mathoverflow.net/questions/105577/… The same tricks should work, you can do a 3-descent via 3-isogeny and a 4-descent by brute force (assuming GRH will help). Those descents can be patched together with the TwelveDescent command in magma to give you 12-covers of the elliptic curve on which to search for points. $\endgroup$ Sep 3, 2014 at 23:58


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.