Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, ℤ[ζ _p] the ring of integers in the pth cyclotomic field, C_{f, p} : y^p = f(x) the corresponding superelliptic curve and J(C _{f, p}) its jacobian. Assuming that either n = p + 1 or p does not divide n(n - 1), we prove that the ring of all endomorphisms of J(C _{f, p}) coincides with ℤ[ζ_p]. The same is true if n = 4, the Galois group of f(x) is the full symmetric group S₄ and K contains a primitive pth root of unity.