of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to $\operatorname{End}_{\overline{K}}(E)$*, defined as follows:

Let $F$ be a reduced ${\mathbb{Q}}_\ell$-algebra of degree $2$ and let ${\mathcal{A}}_\ell$ be a ${\mathbb{Z}}_\ell$-order in $F$. The *Cartan subgroup* corresponding to ${\mathcal{A}}_\ell$ is the group of units of ${\mathcal{A}}_\ell$, which we embed in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ by fixing a ${\mathbb{Z}}_\ell$-basis of ${\mathcal{A}}_\ell$ and considering the left multiplication action of ${\mathcal{A}}_\ell^\times$. If ${\mathcal{A}}$ is an order in an imaginary quadratic number field, the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$ is defined by taking ${\mathcal{A}}_\ell={\mathcal{A}}\otimes {\mathbb{Z}}_\ell$ in the above.

More precisely, when $E/K$ is an elliptic curve with CM, the image of the $\ell$-adic torsion representation $\tau_{\ell^\infty}$ is always contained (up to conjugacy in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$) in the normaliser of the Cartan subgroup corresponding to $\operatorname{End}_{\overline{K}}(E)$, and is contained in the Cartan subgroup itself if and only if the complex multiplication is defined over the base field $K$.

In order to have a practical representation of Cartan subgroups, we recall the following definition from [@MR3690236]:

\[def:CartanParameters\] Let $C$ be a Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$. We say that $(\gamma, \delta) \in {\mathbb{Z}}_\ell^2$ are *parameters for $C$* if $C$ is conjugated in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ to the subgroup $$\label{nfaa}
\left\{ \begin{pmatrix}
x & \delta y \\ y & x+\gamma 