Non-isogenous superelliptic jacobians II

Let \(\ell \) $\ell$ be an odd prime and K a field of characteristic different from \(\ell \) $\ell$ . Let \(\overline{K}\) $\overline{K}$ be an algebraic closure of K. Assume that K contains a primitive \(\ell \hbox {th}\) $\ell \text{th}$ root of unity. Let \(n \ne \ell \) $n\ne \ell$ 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 \(C_{f,\ell }:y^{\ell }=f(x)\) $C_{f,\ell }:y^{\ell }=f(x)$ and \(C_{h,\ell }:y^{\ell }=h(x)\) $C_{h,\ell }:y^{\ell }=h(x)$ of genus \((n-1)(\ell -1)/2\) $(n-1)(\ell -1)/2$ , and their jacobians \(J^{(f,\ell )}\) $J^{(f,\ell )}$ and \(J^{(h,\ell )}\) $J^{(h,\ell )}$ , which are \((n-1)(\ell -1)/2\) $(n-1)(\ell -1)/2$ -dimensional abelian varieties over \(\bar{K}\) $\overline{K}$ . Suppose that one of the polynomials is irreducible and the other reducible over K. We prove that if \(J^{(f,\ell )}\) $J^{(f,\ell )}$ and \(J^{(h,\ell )}\) $J^{(h,\ell )}$ are isogenous over \(\bar{K}\) $\overline{K}$ then both endomorphism algebras \(\textrm{End}^{0}(J^{(f,\ell )})\) ${\text{End}}^0(J^{(f,\ell )})$ and \(\textrm{End}^{0}(J^{(h,\ell )})\) ${\text{End}}^0(J^{(h,\ell )})$ contain an invertible element of multiplicative order n.