The modularity theorem reveals the relationship between elliptic curves and modular forms. Is there a series of steps or an algorithm such that we can obtain the corresponding modular form, when given an elliptic curve defined over the field of rational numbers?
2026-03-25 04:40:37.1774413637
the corresponding modular form of an elliptic curve
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Yes.
(a) You compute the conductor of your elliptic curve (say using Tate's algorithm, or some variant).
(b) You then compute all the weight two Hecke eigenforms at that level (using modular symbols).
(c) You determine how many $q$-expansion coefficients you need to distinguish the different eigenforms. (Either by inspection, or theoretically, using the Sturm bound.) This gives a finite number of $a_p$'s that you need to compute to recognize your desired eigenform.
(d) You compute sufficiently many $a_p$'s for your elliptic curve to determine which eigenform it matches with.
In practice, unless your elliptic curve has an absolutely enormous conductor, this work will have already been done (most likely by Cremona), and you can just look at the existing tables (now incorporated into the $L$-functions and modular forms database here).