Explicit points on the Legendre curve

We study the elliptic curve E given by ^y2 = x(x + 1)(x + t) over the rational function field k(t) and its extensions _{K d} = k(_μd, ^t1/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 Sh{cyrillic}(E/_{K d}). When k has characteristic 0, we show that E has rank 0 over _{K d} for all d.