How do people compute the $j$-invariant of an elliptic curve $E$ over $\mathbb Q$ from the associated modular form $f=\sum_{n=1}^{\infty} a_nq^n$? In other words, how to compute (at least giving some estimates) $j(E)$ using $a_n$?
2026-03-27 01:47:53.1774576073
How to know the j-invariant of the modular elliptic curve from the modular form?
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This is not a well-defined question, because the modular form $f$ corresponds to an isogeny class of elliptic curves, not a single elliptic curve; and the elliptic curves in the isogeny class can have different $j$-invariants.
However, if you ask for the $j$-invariant of some elliptic curve in the isogeny class, then this is a very well-studied problem, and there is an effective way of doing so using the period lattice. This is all described comprehensively and beautifully in John Cremona's book Algorithms for modular elliptic curves (available for free online here); the algorithm you're after is described in section 2.14.