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This paper contains a complete study of superspecial curves of genus $4$ incharacteristic $p\le 7$. We prove that there does not exist a superspecialcurve of genus $4$ in characteristic $7$. This is a negative answer to thegenus $4$ case of the problem proposed by Ekedahl [9] in 1987. This implies thenon-existence of maximal curve of genus $4$ over $\mathbb{F}_{49}$, whichupdates the table at {\tt manypoints.org}. We give an algorithm to enumeratesuperspecial nonhyperelliptic curves in arbitrary $p \ge 5$, and for $p\le 7$we excute it with our implementation on a computer algebra system Magma. Ourresult in $p=5$ re-proves the uniqueness of maximal curves of genus $4$ over$\mathbb{F}_{25}$, see [11] for the original theoretical proof. In Appendix, wepresent a general method determining Hasse-Witt matrices of curves which arecomplete intersections.

Superspecial curves of genus $4$ in small characteristic