This code was used in the article "On periods of elliptic curves" by Daniel Barrera and Juan-Pablo Llerena.
The objective of this code was to study Expectation 8.2 of the aformentioned article. This expectatoin relates the p-adic valuation of a j-invariant of a rational elliptic curve, when p is a prime for which E has split multiplicative reduction.
This code also allowed us to study the more general question: "Let E be a rational elliptic curve of conductor N, let p be a prime such that p divides N and j_E be the j-invariant of E. If ell > 5 is a prime such that ell|p-1 and it is coprime to the modular degree of E, then (ord_p(j_E),ell) = 1".
The code was runned on all the elliptic curves in the cremona database version 2022-10-13: https://zenodo.org/records/7194436
There are 3 files and 1 folder in this project:
Readme.md: This is the file you are currently reading, and contains the general description of this project, it also contains a small explanation of the code.
main.sage: This is the code that was used in the article.
main_2.sage: This is a more basic version of main.sage used primarly for checking one elliptic curve at a time instead of all at once.
alldelphi: For simplification purposes (and following cremona's license) we copied, as is and without modification, the folder named alldelphi of cremona's data base: https://zenodo.org/records/7194436. This folder contains the minimal Wiestrass equation and the j-invariant of all the elliptic curves in the cremona database.
The code used on the article cannot be run with online interpreters (e.g. Cocalc). The reason being that we are unaware if there is a way to upload files into Cocalc which will allow SageMath to have acces to the list of elliptic curves in the cremona database. Nevertheless, the main_2.sage can be used in online interpreters, by changing the variable 'name_of_elliptic_curve'.