compute the equation for elliptic curves from lattices

asked 2022-04-03 05:25:30 +0100

Lee gravatar image

updated 2022-04-05 02:55:06 +0100

Given a lattice in $\mathbb{C}$, is there commands in Sage or Pari to compute the equation of the elliptic curves associated to the lattice? or in other words, compute $c_4$ and $c_6$ of the lattice?

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1

How exactly the lattice is given? Please provide an example.

Max Alekseyev gravatar imageMax Alekseyev ( 2022-04-03 16:04:52 +0100 )edit

For example, given a basis

[1.4752770112052821796 - 0.6182637476411826238*i,
 1.6774834051116878106 - 2.205022990784728253*i,]

of the lattice. How can I compute the equation of $\mathbb{C}/\Lambda$?

Lee gravatar imageLee ( 2022-04-05 02:53:16 +0100 )edit

You'd need to compute Eisenstein series as explained at https://math.stackexchange.com/q/1291245

Max Alekseyev gravatar imageMax Alekseyev ( 2022-04-05 05:51:48 +0100 )edit

Thanks a lot for the anwser ! So there is no program or command in Sage or Pari to do that directly, right?

Lee gravatar imageLee ( 2022-04-06 04:20:03 +0100 )edit

There is no single command for that, but related functionality is present in Sage and Pari -- e.g., see answers at https://ask.sagemath.org/question/51409/

Max Alekseyev gravatar imageMax Alekseyev ( 2022-04-06 05:53:06 +0100 )edit
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