On the existence of superspecial nonhyperelliptic curves of genus $4$

A curve over a perfect field $K$ of characteristic $p>0$ is said to be superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves over the algebraic closure $\bar{K}$. In recent years, isomorphism classes of superspecial nonhyperelliptic curves of genus $4$ over finite fields in small characteristic have been enumerated. In particular, the non-existence of superspecial curves of genus $4$ in characteristic $p=7$ was proved. In this note, we give an elementary proof of the existence of superspecial nonhyperelliptic curves of genus $4$ for infinitely many primes $p$. Specifically, we prove that the variety $C_p : x^3+y^3+w^3= 2 y w + z^2 = 0$intheprojective$3$-spacewith$p > 2$isasuperspecialcurveofgenus$4$ifandonlyif$p \equiv 2 \pmod{3}$\mathrm{.}Ourcomputationalresultsshowthat$C_p$with$p \equiv 2 \pmod 3$aremaximalcurvesover$$\mathbb{F}$_{p^2}$ for all $3\le p\le 269$.