  For $p$ an odd prime number, $q_{0}=p^{t}$, and $q=p^{2t-1}$, let
$\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$ be the nonsingular model of $$
Y^{q}-Y=X^{q_{0}}(X^{q}-X). $$ In the present work, the number of
$\mathbb{F}_{q^{n}}$-rational points and the full automorphism group of
$\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$ are determined. In addition, the
L-polynomial of this curve is provided, and the number of
$\mathbb{F}_{q^{n}}$-rational points on the Jacobian
$J_{\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}}$ is used to construct \'{e}tale
covers of $\mathcal{X}_{\mathcal{G}_{\mathcal{S}}}$, some with many rational
points.
