The endomorphism rings of jacobians of cyclic covers of the projective line

Suppose that K is a field of characteristic zero, K_a is its algebraic closure, and that f(x) ∈ K[x] is an irreducible polynomial of degree n ≥ 5, whose Galois group coincides either with the full symmetric group A_n or with the alternating group A_n. Let p be an odd prime, Z[ζ_p] the ring of integers in the pth cyclotomic field Q(ζ_p). Suppose that C is the smooth projective model of the affine curve y^p = f(x) and J(C) is the jacobian of C. We prove that the ring End(J((C)) of K_a-endomorphisms of J(C) is canonically isomorphic to Z[ζ_P].