Code for the article: On the sum of fourth powers in arithmetic progression

This git repository contains files related to the computational part of the article: On the sum of fourth powers in arithmetic progression. It contains documented Sage code for working with Frey curves, Q-curves and Frey-Q-curves as well as the corresponding newforms. Furthermore the file computations.sage gives all the details for the computations done in the mentioned article. It consists of code snippets combined with explanatory notes to make the computations insightful.

Contents

• calculations.rst A structured file giving all the computations needed for the article accompanied by explanatory notes. This file explains in detail what computations have to be done, the input that has to be provided to compute this and the corresponding output. The output can be automatically verified using Sage's doc-test system (see How to use)

• src A folder containing code written for Sage to perform various computations on Frey curves, Q-curves and Diophantine equations. An effort has been made to document the code according to the Sage guidelines for developers.

• load.sage A script to load all the Sage code into Sage.

• save_nfs.m Some MAGMA code to save newforms to the file format specified in src/modular_forms/newform wrapper.sage.

• generate_nfs.m A magma script that will compute the newforms used in the paper and save them to the files used by calculations.rst.

• README.md This file

Requirements

To use the code the user needs an installation of SageMath version 8.4 or newer. The code was last tested on version 8.7.

The code has an option to compute newforms using MAGMA. For this one needs an installation of MAGMA. The code has been tested on version 2.24-8 so that version or a newer one is recommended. The installation should be set up such that it is accessible through SageMath's MAGMA interface.

If you want to test the computations in _calculations.rst_make sure you have an installation of MAGMA as described before. Furthermore you need a directory in this directory called tmp that contains the files E1.nfs and E2.nfs containing the newform data needed for the computations. These files can be generated using the generate_nfs.m magma script. This script can be executed using the load command from within a magma terminal or directly by opening a terminal inside this directory and typing

>> magma generate_nfs.m

Note that generating these files will take a long time (about a week) and is therefore not recommended. Pre-computed versions of these files are available upon request.

How to use

Checking the computations

The file calculations.rst contains all the computations needed for the paper. The file is written in reStructuredText and can be opened with any text editor. The file contains fragments of code, written as the input on the Sage command line followed by the corresponding output, if any. Some text surrounding these fragments explains what is computed in each part.

All the output in the document can be checked using SageMath's doc-test system. Simply open a terminal in the current directory and type

>> sage -t -T 0 -m 0 calculations.rst

The additional arguments are required, since the computations exceeds the standard time and memory limits of Sage's doc-tests.

After some hours the doc-test should print "All tests passed" and finish, in which case all the output in the document has been verified.

Using the code

In order to use the Sage code and work with Frey curves and other tools developed for this paper, start a Sage terminal in this directory and type

sage: load('load.sage')

which will load all the code from the src directory. Note that the paths in load.sage are relative and thus require the sage terminal to be in this directory to work.