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.

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    $\begingroup$ 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 '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. $\endgroup$ Aug 29 '14 at 17:35
  • $\begingroup$ Try ask.sagemath.org/questions $\endgroup$
    – GH from MO
    Aug 29 '14 at 17:42
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    $\begingroup$ 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 '14 at 1:51
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    $\begingroup$ 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 '14 at 23:58

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