Nonexistence of maximal curves of genus five over ${\text{F}}_{64}$

We show that there is no maximal curve of genus five over ${\text{F}}_{64}$. As a consequence, $N_{64}(5)=140$, and the genus spectrum of maximal curves over ${\text{F}}_{64}$ is determined. The proof uses the vanishing of the third iterate of the Cartier operator. We prove that a nonhyperelliptic curve of genus five in characteristic two satisfying this condition is nontrigonal. Its canonical theta characteristic defines a separable cover of degree four with one geometric branch value. The two possible ramification types give either a rational subcanonical point or a point bound obtained from the cubic resolvent. Both cases exclude maximality over ${\text{F}}_{64}$.