Some Sextics of Genera Five and Seven Attaining the Serre Bound

Abstract

We define two families of sextics. By computer search on one family, we find new curves of genus 5 attaining the Hasse–Weil–Serre bound over \(\mathbb {F}_{71}\), \(\mathbb {F}_{191}\) and \(\mathbb {F}_{11^5}\), and we update 3 entries of genus 5 in manYPoints.org. Among another family, we find new curves of genus 7 attaining the Hasse–Weil–Serre bound over \(\mathbb {F}_{p^3}\) for some primes p. We determine the precise condition on the finite field over which the sextics attain the Hasse–Weil–Serre bound.