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 P4. We present and explain algorithms we used to determine, up to isomorphism over F2, all genus 5 curves defined over F2, and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over F2 of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over F2 weighted by the size of their F2-automorphism groups.