Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A in operatorname{M}_{2g}(mathbb{Z})$ such that each Tate module $T_ell X$ has a $mathbb{Z}_ell$-basis on which the action of $u$ is given by $A$.
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