  In this paper we study the field of definition of abelian subvarieties
$B\subset A_{\overline{K}}$ for an abelian variety $A$ over a field $K$ of
characteristic $0$. We show that, provided that no isotypic component of
$A_{\overline{K}}$ is simple, there are infinitely many abelian subvarieties of
$A_{\overline{K}}$ with field of definition $K_A$, the field of definition of
the endomorphisms of $A_{\overline{K}}$. This result combined with earlier work
of R\'emond gives an explicit maximum for the minimal degree of a field
extension over which an abelian subvariety of $A_{\overline{K}}$ is defined
with varying $A$ of fixed dimension and $K$ of characteristic $0$.
