An ${\mathbb{F}}_{p^2}$-maximal Wiman's sextic and its automorphisms

In 1895 Wiman introduced a Riemann surface $\mathcal{W}$ of genus $6$ over the complex field $\mathbb{C}$ defined by the homogeneous equation $\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2{Y}^2{Z}^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\mathcal{W}$ was previously studied as a curve defined over a finite field ${\mathbb{F}}_{p^2}$ where $p$ is a prime, and necessary and sufficient conditions for its maximality over ${\mathbb{F}}_{p^2}$ were obtained. In this paper we first show that the result of Wiman concerning the automorphism group of $\mathcal{W}$ holds also over an algebraically closed field $\mathbb{K}$ of positive characteristic $p$, provided that $p\ge 7$. For $p=2,3$ the polynomial $X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2$isnotirreducibleover$$\mathbb{K}$$,whilefor$p=5$thecurve$$\mathcal{W}$$isrationaland$Aut($\mathcal{W}$) \cong PGL(2,$\mathbb{K}$)$. We also show that the ${\mathbb{F}}_{{19}^2}$-maximal Wiman’s sextic $\mathcal{W}$ is not Galois covered by the Hermitian curve ${\mathcal{H}}_{19}$ over ${\mathbb{F}}_{{19}^2}$.