Description

This repository contains Magma code for reconstruction and isomorphisms of hyperelliptic curves.

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

Prerequisites

You need to have Magma installed on your machine to use this code.

Installation

You can enable the functionality of this package in Magma by attaching the hyperelliptic/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 a line like

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

Usage

Examples that test the routines in this package can be found in the directory examples (a full list of intrinsics is here).

Verbose comments are enabled by

SetVerbose("Hyperelliptic", n);

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

Basic use of the package is as follows.

• Compute Shioda Invariants [J2, J3, J4, J5, J6, J7, J8, J9, J10] of a genus 3 hyperelliptic curve

_<x> := PolynomialRing(Rationals());
H := HyperellipticCurve(x^8+x^6-x^2+x-1);
JI := ShiodaInvariants(H);
JI;
[ -29/14, 0, 158041/230496, 25/87808, 242163085/1084253184, -97577/14751744, -12622697959/330576748544, -2165139797/1942981705728, -750879849570517/62201321006039040 ];

• Conversely, given the first 6 free invariants J2, J3, J4, J5, J6, J7, compute all the possible algebraically compatible invariants J8, J9, J10 (here, there is only one solution, the one we start from);

JIs := ShiodaAlgebraicInvariants(JI[1..6]);

• Reconstruct from these invariants a model, and the geometric automorphism group of the curve.

C, aut := HyperellipticCurveFromShiodaInvariants(JIs[1]);

• Check that H and C are twists.

ShiodaInvariantsEqual(JI, ShiodaInvariants(C));

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