  Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power
$q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q
-t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$
such as the rank of its Mordell--Weil group $E(K)$, the size of its
N\'eron--Tate regulator $\text{Reg}(E)$, and the order of its Tate--Shafarevich
group $III(E)$ (which we prove is finite). These invariants have radically
different behaviors depending on the congruence class of $p$ modulo 6. For
instance $III(E)$ either has trivial $p$-part or is a $p$-group. On the other
hand, we show that the product $|III(E)|\text{Reg}(E)$ has size comparable to
$r^{q/6}$ as $q\to\infty$, regardless of $p\pmod{6}$. Our approach relies on
the BSD conjecture, an explicit expression for the $L$-function of $E$, and a
geometric analysis of the N\'eron model of $E$.
