Let $A/k$ be a simple abelian variety. 


Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?


I don't need $f:B\to A$ to be etale.



Is any simple abelian variety covered by a non-simple abelian variety
No. If $End\_A$ and $End\_B$ are the endomorphism rings then an isogeny $B\to A$ will give an isomorphism $End\_A\otimes\mathbb Q\to End\_B\otimes\mathbb Q$. But the first of these algebras is a division algebra and the second is not.


