Non-isogenous elliptic curves and hyperelliptic jacobians

Let $E_f: y^2=f(x)$ and $E_h: y^2=h(x)$ be elliptic curves over a field
$K$ of characteristic zero that are defined by cubic polynomials
$f(x)$ and $h(x)$ with coefficients in $K$.

Suppose that one of the polynomials is irreducible and the other
reducible.
We prove that if $E_f$ and $E_h$ are isogenous over an algebraic closure
$\bar{K}$ of $K,$ then they both are isogenous
to the elliptic curve $y^2=x^3-1$ over $\bar{K}$.