Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields The curve explicit divisors and relations Descent calculations Minimal regular model local invariants and domination by a product of curves Heights and the visible subgroup The L function and the BSD conjecture Analysis of J[p] and NS(Xd)tor Index of the visible subgroup and the Tate Shafarevich group Monodromy of ℓ torsion and decomposition of the Jacobian. We study the Jacobian J of the smooth projective curve C of genus r 1 with affine model yr = xr 1(x+ 1)(x + t) over the function field Fp(t) when p is prime and r [greater than or equal to] 2 is an integer prime to p. When q is a power of p and d is a positive integer we compute the L function of J over Fq(t1 d) and show that the Birch and Swinnerton Dyer conjecture holds for J over Fq(t1 d). When d is divisible by r and of the form p[nu] + 1 and Kd := Fp([mu]d t1 d) we write down explicit points in J(Kd) show that they generate a subgroup V of rank (r 1)(d 2) whose index in J(Kd) is finite and a power of p and show that the order of the Tate Shafarevich group of J over Kd is [J(Kd) : V ]2. When r > 2 we prove that the new part of J is isogenous over Fp(t) to the square of a simple abelian variety of dimension [phi](r) 2 with endomorphism algebra Z[[mu]r]+. For a prime with pr we prove that J[](L) = {0} for any abelian extension L of Fp(t)