  We give a natural parameterization of the N\'eron-Severi group of a product
$A = E\times E'$ of two elliptic curves in terms of quadratic forms. As an
application, we determine (in the non-CM case) whether $A$ contains a smooth
curve of any fixed genus. We also determine whether $A$ admits a very ample
line bundle of any fixed degree. In particular, we determine which of these
abelian surfaces embed in $\mathbb{P}^4$, i.e. which come from the
Horrocks-Mumford bundle.
