Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
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1$\begingroup$ I know that Jean Kieffer is doing things of that kind members.loria.fr/JKieffer . By the way, I don't understand why people are downvoting this question. At least, they have to explain why downvoting the question (in my opinion). $\endgroup$– Marsault ChabatCommented Jul 31 at 3:57
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Again an early source explaining how this is done in practice is Cremona's book. Specifically section 3.8.
One current implementation for finding the isogeny class of an elliptic curves over a number field is in sage. See here for the documentation and here for the source.
One of the methods is based on this paper by Nicholas Billerey. This is also used in the implementation of ellisomat in Pari/GP. See page 526 in the documentation.
answered Jul 30 at 12:06
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$\begingroup$ I guess that one could add a lot to this answer, but the question is maybe not precise enough to know what to include. $\endgroup$ Commented Jul 30 at 12:09
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$\begingroup$ One possible question I might have is if it should be possible to compute all possible $j$-invariants of isogenic elliptic curves analyticially, to an arbitrary sharp precision, from the traces of Frobenius of the Tate module of a representative. Your answers are very helpful despite my imprecision. $\endgroup$ Commented Jul 30 at 12:17
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$\begingroup$ I will look at all the references you've shared, but if you could direct me more specifically if any of these methods offer such a feature, I'd be grateful $\endgroup$ Commented Jul 30 at 12:18