For
p
p
an odd prime number,
q
0
=
p
t
q_{0}=p^{t}
, and
q
=
p
2
t
−
1
q=p^{2t-1}
, let
X
G
S
\mathcal {X}_{\mathcal {G}_{\mathcal {S}}}
be the nonsingular model of
Y
q
−
Y
=
X
q
0
(
X
q
−
X
)
.
\begin{equation*} Y^{q}-Y=X^{q_{0}}(X^{q}-X). \end{equation*}
In the present work, the number of
F
q
n
\mathbb {F}_{q^{n}}
-rational points and the full automorphism group of
X
G
S
\mathcal {X}_{\mathcal {G}_{\mathcal {S}}}
are determined. In addition, the L-polynomial of this curve is provided, and the number of
F
q
n
\mathbb {F}_{q^{n}}
-rational points on the Jacobian
J
X
G
S
J_{\mathcal {X}_{\mathcal {G}_{\mathcal {S}}}}
is used to construct étale covers of
X
G
S
\mathcal {X}_{\mathcal {G}_{\mathcal {S}}}
, some with many rational points.