Superelliptic jacobians and central simple representations

Let f(x) be a polynomial of degree at least 5 with complex coefficients and without repeated roots. Let p be an odd prime. Suppose that all the coefficients of f(x) lie in a subfield K such that: 1) K contains a primitive p-th root of unity; 2) f(x) is irreducible over K; 3) the Galois group \Gal(f) of f(x) acts doubly transitively on the set of roots of f(x); 4) the index of every maximal subgroup of Gal(f) does not divide deg(f)-1. Then the endomorphism ring of the Jacobian of the superelliptic curve y^p=f(x) is isomorphic to the pth cyclotomic ring for all primes p>deg(f).