Let us say that I have a complex abelian variety $A$, an ample line bundle on $A$, $L$, and an effective divisor $E\in|L|$. It is well known that it exists an isogeny $\varphi:A\rightarrow B$ and a principal polarization $M$ on $B$ such that $L\simeq \varphi^*M$. My question is: can we say something about the divisor $\varphi(E)$? For example it is possible that the map $\varphi$ restricted to $E$ has degree 1? I do not think this may happen, but I do not know how to see it...\
is there any chance that $\varphi(E)$ is a $\Theta$-divisor in $B$?