This is the GitHub repository containing code related to the paper Explicit 7-torsion in the Tate-Shafarevich groups of genus 2 Jacobians by Sam Frengley arXiv:2405.11693.

Most of the code in this repository is written in Magma, some verifications are performed in SageMath. We have tested on Magma version 2.27-8.

The intrinsics in src

The main algorithms described in the paper are implemented in the files in the directory src. In particular, the user should be aware of the intrinsic KleinQuarticTwists which takes as input a degree 6 polynomial defining a genus 2 curve with RM by $\mathcal{O}_D$ in which 7 splits and outputs four twists of the Klein quartic which are conjecturally the ones associated to this curve.

Data

The data directory includes (for example) the results of Theorem 1.1 and 1.2. We include several examples of twists of the Klein quartic associated to genus 2 curves.

Verifications

The verify directory provides proofs of claims made in the paper. Namely, we provide code to prove:

• Theorems 1.1 and 1.5,
• Proposition 4.4, and
• Proposition 5.2.