Computing binary curves of genus five

TL;DR

This paper develops algorithms to classify all genus 5 curves over the finite field _2, categorizing them by their geometric types and analyzing their Jacobian isogeny classes, automorphisms, and Newton polygons.

Contribution

The paper introduces specific algorithms for enumerating all genus 5 curves over _2 up to isomorphism, distinguished by their geometric types and Jacobian properties.

Findings

Classified all genus 5 curves over _2 by type.

Computed the number of curves weighted by automorphism group size.

Analyzed isogeny classes and Newton polygons of Jacobians.

Abstract

Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in $\mathbb{P}^4$. We present and explain algorithms we used to determine, up to isomorphism over $\mathbb{F}_2$, all genus 5 curves defined over $\mathbb{F}_2$, and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over $\mathbb{F}_2$ of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over $\mathbb{F}_2$ weighted by the size of their $\mathbb{F}_2$-automorphism groups.