Computing binary curves of genus five

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.