Frobenius endomorphism of Abelian varieties

Question

Let $A$ be an abelian variety over $\mathbb{F}_q$ with $q=p^n$, such that all of it's endomorphisms are defined over $\mathbb{F}_q$. Then $End(A)\otimes \mathbb{Q}$ is of Albert type (IV) with center $L$ a CM-field. Let $T_p(A)$ be it's p-divisible group. Due to the Tate-conjecture $End(A)\otimes \mathbb{Z}_p\xrightarrow{\sim} End(T_p(A))$ are isomorphic. But on one hand the Frobenius endomorphism $\pi_A$ of $A$ is in $L$ and on the other hand given the description of $End(T_p(A)) \otimes \mathbb{Q}_p$ from Dieudonne theory, $\pi_{A}$ can not be central in $End(T_p(A)) \otimes \mathbb{Q}_p$. So what am I missing here?