  Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K=
\mathbb{F}_q(t)$. For any integer $d\geq 2$ coprime to $q$, consider the
elliptic curve $E_d$ over $K$ defined by $y^2=x(x^2+t^{2d} x-4t^{2d})$. We show
that the rank of the Mordell--Weil group $E_d(K)$ is unbounded as $d$ varies.
The curve $E_d$ satisfies the BSD conjecture, so that its rank equals the order
of vanishing of its $L$-function at the central point. We provide an explicit
expression for the $L$-function of $E_d$, and use it to study this order of
vanishing in terms of $d$.
