So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed components, $L^{\otimes 2}$ is very ample, and for $d_1=2$ we have base point freeness.

What about the case where $d_1=1$? Certainly it's not always base point free, for instance, in the case of a principal polarization. But what about nonprincipal polarizations? Are there any known criteria for determining if they are base point free or not? And if there is a base locus, for determining it (which presumably requires knowledge of $L$, not just the type of $L$)?

The only result I can find in this direction is that for an abelian surface with a $(1,d)$ polarization, we have no base points for $d\geq 3$ and for $d=2$ we have exactly 4 base points, but what about higher dimensional $X$?

EDIT: As pointed out below, I want to assume that $X$ is simple.