Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding Weierstrass $P$ function) but I don't know how to get it explicitly from the elliptic invaiants $g_2$ and $g_3$.
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1The converse question is here: https://math.stackexchange.com/questions/1291245 – Watson Oct 26 '18 at 14:33
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See also https://math.stackexchange.com/questions/309882/ – Watson Jun 07 '21 at 16:51
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The generators of the lattice are given by line integrals of the differential dx/y. See Silverman's "The Arithmetic of Elliptic Curves", Chapter 6, in particular Proposition 5.2.
If you want to know how this is actually done, I'd suggest Cohen's "A Course in Computational Algebraic Number Theory", Chapter 7, in particular Algorithm 7.4.7.
Álvaro Lozano-Robledo
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