Rational Tate module
= Rational Tate module
{title2=$V_\ell(A)=T_\ell(A)\otimes_{\mathbb Z_\ell}\mathbb Q_\ell$}
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.