  We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational
function field k(t) and its extensions K_d=k(\mu_d,t^{1/d}). When k is finite
of characteristic p and d=p^f+1, we write down explicit points on E and show by
elementary arguments that they generate a subgroup V_d of rank d-2 and of
finite index in E(K_d). Using more sophisticated methods, we then show that the
Birch and Swinnerton-Dyer conjecture holds for E over K_d, and we relate the
index of V_d in E(K_d) to the order of the Tate-Shafarevich group \sha(E/K_d).
When k has characteristic 0, we show that E has rank 0 over K_d for all d.
