Non-isogenous superelliptic jacobians II

Let ℓ be an odd prime and K a field of characteristic different from ℓ. Let K¯ be an algebraic closure of K. Assume that K contains a primitive ℓth root of unity. Let n≠ℓ be another odd prime. Let f(x) and h(x) be degree n polynomials with coefficients in K and without repeated roots. Let us consider superelliptic curves Cf,ℓ:yℓ=f(x) and Ch,ℓ:yℓ=h(x) of genus (n-1)(ℓ-1)/2, and their jacobians J(f,ℓ) and J(h,ℓ), which are (n-1)(ℓ-1)/2-dimensional abelian varieties over K¯. Suppose that one of the polynomials is irreducible and the other reducible over K. We prove that if J(f,ℓ) and J(h,ℓ) are isogenous over K¯ then both endomorphism algebras End0(J(f,ℓ)) and End0(J(h,ℓ)) contain an invertible element of multiplicative order n.