Superspecial trigonal curves of genus $5$

TL;DR

This paper introduces an algorithm to enumerate superspecial trigonal curves of genus 5 over finite fields, implemented in Magma, covering various prime fields and extension degrees.

Contribution

It presents a novel algorithm for enumerating superspecial trigonal curves of genus 5 over finite fields, with computational results for specific primes and extension degrees.

Findings

Enumerated superspecial trigonal genus 5 curves over finite fields for p ≤ 7 and odd a for p ≤ 13.

Implemented the enumeration algorithm in Magma.

Provided explicit classifications over various finite fields.

Abstract

This paper provides an algorithm enumerating superspecial trigonal curves of genus $5$ over finite fields. Executing the algorithm over a computer algebra system Magma, we enumerate them over finite fields $\mathbb{F}_{p^a}$ for any natural number $a$ if $p \leq 7$ and for odd $a$ if $p \leq 13$.