Maximal curves of genus 5 over finite fields

A maximal curve over a finite field ${\mathbb{F}}_q$ is a curve whose number of points reaches the upper Hasse-Weil-Serre bound. We define the discriminant of ${\mathbb{F}}_q$ as $d({\mathbb{F}}_q):=\lfloor 2\sqrt{q}{\rfloor }^2-4q$, which arises as the discriminant of the characteristic polynomial of the Frobenius for a maximal elliptic curve defined over ${\mathbb{F}}_q$. In this article we investigate the existence of a maximal curve of genus $5$ defined over a finite field ${\mathbb{F}}_q$ of discriminant $-19$. Using the knowledge on the automorphism group of such a curve, we prove that such curve does not exist when $q\equiv 2,3,4\mathrm{m}\mathrm{o}\mathrm{d}5$. In the case $q\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}5$ we give models of the potential maximal curve. Finally, for the case $q\equiv 05$, we prove that such a curve might exist only for $q=5^7$.