  Let $E$ be the elliptic curve $y^2=x(x+1)(x+t)$ over the field $\Fp(t)$ where
$p$ is an odd prime. We study the arithmetic of $E$ over extensions
$\Fq(t^{1/d})$ where $q$ is a power of $p$ and $d$ is an integer prime to $p$.
The rank of $E$ is given in terms of an elementary property of the subgroup of
$(\Z/d\Z)^\times$ generated by $p$. We show that for many values of $d$ the
rank is large. For example, if $d$ divides $2(p^f-1)$ and $2(p^f-1)/d$ is odd,
then the rank is at least $d/2$. When $d=2(p^f-1)$, we exhibit explicit points
generating a subgroup of $E(\Fq(t^{1/d}))$ of finite index in the "2-new" part,
and we bound the index as well as the order of the "2-new" part of the
Tate-Shafarevich group.
