Non-isogenous elliptic curves and hyperelliptic jacobians II

Let $K$ be a field of characteristic different from $2$, $\bar{K}$ its algebraic closure. Let $n\ge 3$ be an odd integer. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)\mathrm{/}2$ hyperelliptic curves $C_f:y^2=f(x)$ and $C_h: y^2=h(x)$,andtheirjacobians$J(C_f)$and$J(C_h)$,whichare$(n-1)/2$-dimensionalabelianvarietiesdefinedover$K$. Suppose that one of the polynomials is irreducible and the other splits completely over $K$. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\bar{K}$ then there is an (odd) prime $\mathrm{\ell }$ dividing $n$ such that the endomorphism algebras of both $J(C_f)$ and $J(C_h)$ contain a subfield that is isomorphic to the field of $\mathrm{\ell }$th roots of $1$.