A Census of Curves of Genus 6 over $\mathbb{F}_2$

by Yongyuan Huang, Kiran S. Kedlaya, and Jun Bo Lau

This repository contains the code (written in SageMath and Magma) and data used to build a census of curves of genus 6 over the finite field $\mathbb{F}_2$. This is based primarily on Kedlaya's paper "The relative class number one problem for function fields, III" and its associated code repository. Some findings:

• There are 72227 isomorphism classes and 38327 isogeny classes of curves of genus 6 over $\mathbb{F}_2$.
• # $\mathcal{M}_6(\mathbb{F}_2)= 68615$.

The directory Census contains the code and data for each stratum in the Brill-Noether statification of the moduli stack $\mathcal{M}_6$ of curves of genus 6. Each stratum is represented by a single subdirectory with its own README:

• Hyperelliptic curves (Census/hyperelliptic)
• Trigonal curves of Maroni invariant 0 (Census/trigonal_maroni_0)
• Trigonal curves of Maroni invariant 2 (Census/trigonal_maroni_2)
• Bielliptic curves (Census/bielliptic)
• Smooth plane quintic curves (Census/plane_quintic)
• All others (Census/generic)

The subdirectory Census/Shared includes some utility code taken from https://github.com/kedlaya/same-class-number.

Some additional utility files can be found in this directory:

• assemble_data.sage: compiles a Sage list of representatives of isomorphism classes.
• data-to-txtfile.sage: compiles a list of isogeny classes of Jacobians indexed by their LMFDB labels.
• query_data.sage: provides some examples of how to query the data.
• test_nonisomorphism.sage: verifies that all of the curves are pairwise nonisomorphic (even across Brill-Noether strata).

Our data is also available via the table of isogeny classes of abelian varieties over finite fields in the LMFDB.

We thank the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation and Boston University's Department of Mathematics and Statistics for providing computational resources for this project.