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For any smooth Hurwitz curve $\mathcal{H}_n: \, XY^n+YZ^n+X^nZ=0$ over thefinite field $\mathbb{F}_{p}$, an explict description of its Weierstrass pointsfor the morphism of lines is presented. As a consequence, the full automorphismgroup ${\rm Aut}(\mathcal{H}_n)$, as well as the genera of all Galois subcoversof $\mathcal{H}_n$, with $n\neq 3, p^r$, are computed. Finally, a question byF. Torres on plane non nonsingular maximal curves is answered.

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines