Does anybody have a reference for the following fact?

All abelian varieties with complex multiplication and same CM type are isogenous over $\overline{\mathbb{Q}}$?

Here *abelian variety with complex multiplication* means an abelian variety A over $\overline{\mathbb{Q}}$ such that there exists a CM number field $K$ of degree twice the dimension of A and an embedding of $K$ into $End(A) \otimes \mathbb{Q}$. The CM type is obtained by looking at the action of $K$ into $H^0(A, \Omega^1_A)$.