Let $A/\mathbb{Q}$ be an abelian variety which is not simple over $\overline{\mathbb{Q}}$. Let $\phi$ be an isogeny (defined over some number field $K$) from $A$ to its geometrically simple components. What can one say about $K$?
I think $K$ should contain $End(A_{\overline{\mathbb{Q}}}) \otimes \mathbb{Q}$ but not sure what else can be said.