  For an abelian variety $A$ over a number field we study bounds depending only
on the dimension of $A$ for the minimal degree $d(A)$ of a field extension over
which $A$ acquires semi-stable reduction. We first compute $d(A)$ in terms of
the cardinalities of the finite monodromy groups of $A$ which leads to a bound
on $d(A)$ in terms of the classical Minkowski bound. We then show this bound is
tight up to its $2$-part by considering $p$-adic coverings of the local points
of a universal abelian scheme.
