Rational Tate module ($V_{\ell }(A)=T_{\ell }(A){\otimes }_{{\mathbb{Z}}_{\ell }}{\mathbb{Q}}_{\ell }$)

Tensoring a Tate module with the P-adic numbers gives its rational Tate module. For a CM elliptic curve with base-defined multiplication by $K$, it is free of rank one over $K\otimes {\mathbb{Q}}_{\ell }$, including when this algebra is a product of two fields. In the split case the two distinct roots of a nonrational CM endomorphism each have a one-dimensional eigenspace.