This repository gives a set of algorithms for calculating with Picard curves, as well as furnishing a small database of Picard curves having good reduction outside two primes in { 2, 3, 5, 7 }.
An installation of both Magma and SageMath is required to run all of the code. You will also benefit from installing a recent version of MCLF/mclf, but this is not mandatory.
The subdirectory picard_curves/magma/ includes code that can be run purely within Magma. You can load all the Magma specific files by attaching the picard_curves/magma/spec file with AttachSpec. For example, if you start your session of Magma inside the git directory, you can do this by typing
AttachSpec("picard_curves/magma/spec");
To install the package in SageMath, first clone the repository via
git clone https://github.com/jrsijsling/picard_curves.git
then go to the newly created directory and type
sage -pip install --user --upgrade .
Once the package is updated on GitHub, pulling the new changes and running the same command will update your installation. After this, a new package called picard_curves will be available for import in SageMath, and the functions in the packages can be loaded with
sage: from picard_curves import *
Examples, both in Magma and in SageMath, are given in the directories starting with examples. We refer to the files in these directories for more details. In particular, try starting with examples_magma/curves.m and examples_sage/curves.sage.
The creation of database files in SageMath and the interaction with them is described in the directory database/. Reading and using these databases is described in the file examples_sage/inspect_database.sage.
Finally, some files confirming results in the database are given in the directory paper/.
This package uses code from the Magma package echidna by David Kohel for an implementation of the Dixmier--Ohno invariants.
This implementation is based on the following work. When using this package, please be aware of the work that you are indirectly applying and please cite it.
- Irene Bouw, Angelos Koutsianas, Jeroen Sijsling, and Stefan Wewers, Conductor and discriminant of Picard curves