Potential toric rank (awaiting review)

Let $A$ be an abelian variety over a number field $K$, and let $\mathfrak{p}$ be a prime of $K$. Then after a finite extension $K \subset L$, there is an integral model of $A$ whose reduction is a semiabelian variety, i.e. the extension of an abelian variety by a torus. The potential toric rank of $A$ at $\mathfrak{p}$ is the dimension of that torus.

As a special case, $A$ has potential good reduction if and only if its potential toric rank is 0.