A note on certain superspecial and maximal curves of genus $5$

In this note, we characterize certain maximal superspecial curves of genus $5$ over finite fields. Specifically, we prove that the desingularization $T_p$ of $xy{z}^3+x^5+y^5=0$ is a maximal superspecial trigonal curve of genus $5$ if and only if $p\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}15)$ or $p\equiv 11(\mathrm{m}\mathrm{o}\mathrm{d}15)$. Moreover, we give families of maximal curves of more general type, which include $T_p$.