We study when Hurwitz curves are supersingular. Specifically, we show that the
curve
,
with
and
relatively prime, is supersingular over the finite field
if and only if there
exists an integer
such that
.
If this holds, we prove that it is also true that the curve is maximal over
.
Further, we provide a complete table of supersingular Hurwitz curves of genus less
than 5 for characteristic less than 37.
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