  For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an
invariant and show that the polarization is basepoint free or projectively
normal if the invariant is small. Their result is generalized to higher
syzygies by F. Caucci, that is, the polarization satisfies property $(N_p)$ if
the invariant is small. In this paper, we study a relation between the
invariant and degrees of abelian subvarieties with respect to the polarization.
For abelian threefolds, we give an upper bound of the invariant using degrees
of abelian subvarieties. In particular, we affirmatively answer some questions
on abelian varieties asked by the author, V. Lozovanu and Caucci in the three
dimensional case.
