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Computing generators of a Mordell curve $$y^2 = x^3 - 44275089430000,$$ can be done in Magma by running the following code:

SetClassGroupBounds("GRH"); Generators(EllipticCurve([0,-44275089430000]))

Multiple runs of this code in Magma calculator produces two different outcomes:

[] false false

or

[ (18748450031360702684179393606032922472692419624480580773326199534256/3396838\ 01416243388455124782355212643880890159427986479577965225 : 6967657247521859894337760812353908599119616220367281714549084326793359797075167\ 8246596820708687007596/62605466439213150762495390467855187870102299492791336241\ 99210595756050841608942926666398752875 : 1) ] true true

The former result means that Magma failed to compute the rank (and generators) of the corresponding Mordell-Weil group, while the latter means it succeeded in that. The discrepancy in possible results seems to be explained by the random choices made during computations. The results can be deterministically reproduced by fixing the random seed as SetSeed(3); or SetSeed(0);, respectively.

So, the failure in computing generators for a given curve does not necessarily mean that Magma cannot do that, but may be just a result of particular "unlucky" random choices. This leads to practical questions:

Q: What are the chances of being just "unlucky" with computation of generators? What is the probability of remaining "unlucky" in a series of $n$ repeated computations (with different random seeds) of the generators for the same curve?

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  • $\begingroup$ Random algorithms use a seed, which is just a long number. If you want a reproducible result, you can set the seed at the beginning of the code. If you don't, it still tells you what the seed was, so that you can reproduce the computation. No, that doesn't answer your question. I knew that. $\endgroup$ Commented Jul 30, 2024 at 18:51
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    $\begingroup$ The Magma calculator has a limit of 2 minutes. So the failure just means that Magma can not find the generators in that time. $\endgroup$ Commented Jul 30, 2024 at 19:46
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    $\begingroup$ You shouldn't run a random algorithm like this several times with the same arguments; instead you should tweak the optional arguments and finer methods. For instance MordellWeilRankBounds has an argument Effort that you can increase. Definitely you should use analytic methods, like Heegner points, for curves which end up having rank 1. $\endgroup$ Commented Jul 30, 2024 at 20:41
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    $\begingroup$ Actually @OlegEroshkin, I have the full version of Magma on my laptop. In a quick trial of 20 attempts, it failed 13 out of 20 times. So the question is valid for the full version as well. $\endgroup$ Commented Jul 30, 2024 at 20:48
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    $\begingroup$ @OlegEroshkin: It does not hit the time limit as it fails (when it fails) in much less than 120 seconds. $\endgroup$ Commented Jul 30, 2024 at 21:24

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