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Does there exist an algorithm or something of the sort to reverse-engineer a curve from its modular form (weight two eigenform with complex coefficients)? I am aware that sometimes there isn’t a unique elliptic curve paired with a given modular form, but is there way to get just one?

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    $\begingroup$ Yes, this is possible. I think this can be done by computing the period lattice, see chapter 2 here homepages.warwick.ac.uk/~masgaj/book/fulltext/index.html $\endgroup$
    – Will Sawin
    Nov 11, 2022 at 1:33
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    $\begingroup$ I wouldn't call it "reverse engineering" as this is much older than the way to associate a modular form to the elliptic curve. For instance the Antwerp IV tables did this systematically, then Cremona's tables (Will's link above) now on lmfdb extended it. $\endgroup$ Nov 11, 2022 at 14:33

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