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 $y^2=x^36321363052$ 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 offtopic because it is about using the software Sage. Please ask in a Sage forum on question/answer site. $\endgroup$ – Federico Poloni Aug 29 '14 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. $\endgroup$ – Felipe Voloch Aug 29 '14 at 17:35

$\begingroup$ Try ask.sagemath.org/questions $\endgroup$ – GH from MO Aug 29 '14 at 17:42

6$\begingroup$ I think one prefers to use the algebraic method to compute the 2Selmer group, as opposed to the invariant method. In Sage, this might be implemented by Simon in PARI. The 2Selmer 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 $77367184671463906202575142600893705817/4363223875270025353581185386334464$. $\endgroup$ – NAME_IN_CAPS Aug 30 '14 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 3descent via 3isogeny and a 4descent by brute force (assuming GRH will help). Those descents can be patched together with the TwelveDescent command in magma to give you 12covers of the elliptic curve on which to search for points. $\endgroup$ – Jamie Weigandt Sep 3 '14 at 23:58