This is an implementation of two-cover descent for genus 2 curves with a rational Weierstrass point. The theory and formulas are explained in the following paper: https://arxiv.org/abs/2009.10313. The raw data for the results of running this code on a dataset of 7692 curves of Mordell–Weil rank 2 or 3 can be found here: https://github.com/HastD/twocover-results
The main files are twocovers.m and twocover-processor.sage, containing the Magma implementation and a SageMath script that handles data processing and control flow for the algorithm using Sage's interface to Magma. Here is an example of how the script is used:
sage twocover-processor.sage --label 6443.a.6443.1 --output_directory ./results --stages all
An incomplete pure Sage implementation can be found in twocovers.sage. Note that the Sage code only implements geometric constructions, since Sage doesn't have the same facilities for computing Mordell-Weil groups or running the elliptic Chabauty method.
The code that generates the data for the "Examples" section of the paper is in paper-examples.sh. If you're running this on your own system, delete the lines starting with module avail (these are instructions for the BU Shared Computing Cluster, which I used to run the code) and set $outdir to the directory where you want the output files to be written. You'll also first need to run
sage --preparse twocover-processor.sage
to generate the .sage.py file; it's more efficient (and avoids possible I/O errors if running several processes concurrently) to preparse the Sage file once at the outset and then call Sage with the --python option on the generated Python file afterward.
The --index option can be used in place of --label to specify the label at the given index in the supplied list of labels (default data/labels.json; can be set to a different file using the --label_list option). This is convenient for running the code on a large list of curves.
The file data/g2c_curves-r2-w1.json contains the coefficients of the hyperelliptic polynomials of each curve in the aforementioned dataset of 7692 genus 2 curves. If you want to run the code on curves not in this dataset, you will need to create a new JSON file with the coefficients of the curves in an associative array in the following format:
{"label1": [[coefficients of f], [coefficients of h]], "label2": ...}
where the labels will be the inputs to the --label option and the curve has equation y^2 + h(x)*y = f(x). Use the --database option to tell the script where to find the coefficients.