On a remark of Serre

Inspired by a remark of Serre, we extend the search for primes $p$ such that the maximum Hasse bound for the number of points on an elliptic curve over ${\mathbb{F}}_{p^5}$ is not achieved. We then give a list of all $q<{10}^{70}$ such that the Hasse bound is not achieved over ${\mathbb{F}}_q$. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus $2$ and $3$ curves.