jrsijsling/quartic

A Magma package for calculating with smooth plane quartic curves

Description

This repository contains Magma code for reconstructing plane quartic curves from their Dixmier--Ohno invariants and for calculating isomorphisms between plane quartic curves. The latter code only works over the rationals and finite fields. In some non-generic cases, we use an approach due to Andrew Sutherland that involves Groebner bases.

This package is now included by default in Magma-v2.26.

Prerequisites

An installation of Magma.

The algorithms use the Magma algorithm MinimizeReducePlaneQuartic, due to Elsenhans and Stoll, to simplify any output over the rationals. In order for this to work properly, one needs a bug fix of a function at the end of the file magma/package/Geometry/SrfDP/minred.m, which can be found in magma/BugFix.m. Be warned that this new file is attached by default and supersedes its version in Magma.

Installation

You can enable the functionality of this package in Magma by attaching the quartic/magma/spec file with AttachSpec. To make this independent of the directory in which you find yourself, and to active this on startup by default, you may want to indicate the relative path in your ~/.magmarc file, by adding the line

AttachSpec("~/Programs/quartic/magma/spec");

Usage

Examples are given in the directory examples (a full list of intrinsics is here).

Verbose comments are enabled by

SetVerbose("Quartic", n);

where n is either 1 or 2. A higher value gives more comments.

Credits

This package uses code from the Magma package echidna by David Kohel for an implementation of the Dixmier--Ohno invariants.

Citing this code

Please cite the following preprint if this code has been helpful in your research:

Reynald Lercier, Jeroen Sijsling, Christophe Ritzenthaler, Functionalities for genus 2 and 3 curves, arXiv:2102.04372