On the maximality of genus-3 nonhyperelliptic curves of Ciani type

In this paper, we study a Ciani curve $C: x^4 + y^4 + z^4 + rx^2y^2 + sy^2z^2

• tz^2x^2 = 0$inpositivecharacteristic$p \geq 3$\mathrm{.}Wewillshowthatif$C$issuperspecial,thenitsstandardformismaximalorminimalover$$\mathbb{F}${p^2}$withouttakingits$$\mathbb{F}${p^2}$-form.