The Supersingularity of Hurwitz Curves

We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n,\mathrm{\ell }}:X^n{Y}^{\mathrm{\ell }}+Y^n{Z}^{\mathrm{\ell }}+Z^n{X}^{\mathrm{\ell }}=0$, with $n$ and $\mathrm{\ell }$ relatively prime, is supersingular over the finite field ${\mathbb{F}}_p$ if and only if there exists an integer $i$ such that $p^i \equiv -1 \bmod (n^2 - n\ell + \ell^2)$. If this holds, we prove that it is also true that the curve is maximal over ${\mathbb{F}}_{p^{2i}}$. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.