  Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be
relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the
superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the
Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and
Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we
compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we
estimate several arithmetic invariants of $J$ appearing in BSD, including the
rank of the Mordell--Weil group $J(\mathbb F_{r}(t))$, the Faltings height of
$J$, and the Tamagawa numbers of $J$ in terms of the parameters $a,b,q$. For
any $p$ and $r$, we show that for certain $a$ and $b$ depending only on $p$ and
$r$, these Jacobians provide new examples of families of simple abelian
varieties of fixed dimension and with unbounded analytic and algebraic rank as
$q$ varies through powers of $p$. Under a different set of criteria on $a$ and
$b$, we prove that the order of the Tate--Shafarevich group of $J$ grows
quasilinearly in $q$ as $q \to \infty.$
