Cyclic covers of prime power degree, jacobians and endomorphisms

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Suppose that K is a field of characteristic 0, Kₐ is its algebraic closure, p is a prime, q=pʳ is a power prime. Suppose that f(x) ∈ K[x] is a polynomial of degree n > 4 without multiple roots. Let us consider the superelliptic curve C: yq=f(x) and its jacobian J(C). We study the endomorphism algebra End⁰(J(C)) of all Kₐ-endomorphisms of J(C). We prove that End⁰(J(C)) is "as small as possible" if the Galois group of f over K is either the full symmetric group Sₙ or the alternating group Aₙ.