It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal polarization $M$ such that $\phi^*M=L$.

If instead I take a nef but not ample divisor $D$ on $A$, is it always true that $D$ is the pullback of some ample divisor on an abelian subvariety of $A$?

Thanks for help!

It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal polarization $M$ such that $\phi^*M=L$.

If instead I take a nef but not ample divisor $D$ on $A$, is it always true that $D$ is the pullback of some ample divisor on a quotient of $A$?

Thanks for help!