y
\end{pmatrix} : x,y \in {\mathbb{Z}}_{\ell},\; v_{\ell}(x(x+\gamma y)-\delta y^2)=0 \right\}\,.$$ Parameters for $C$ always exist, see [@MR3690236 §2.3].

\[rem:Parameters\] One may always assume that $\gamma,\delta$ are integers. Furthermore, one can always take $\gamma \in \{0,1\}$, and $\gamma=0$ if $\ell \neq 2$.

We also recall the following explicit description of the normaliser of a Cartan subgroup [@MR3690236 Lemma 14]:

\[lem-Norm\] A Cartan subgroup has index 2 in its normaliser. If $C$ is as in , its normaliser $N$ in $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ is the disjoint union of $C$ and $\displaystyle C':=\begin{pmatrix}
1 & \gamma \\ 0 & -1
\end{pmatrix} \cdot C\,.$

Properties of the torsion representation {#sec:PropertiesTorsionRepresentation}
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Torsion representations are studied extensively in the literature; we have in particular the following fundamental theorem of Serre [@Serre], which applies to all elliptic curves (defined over number fields) without complex multiplication:

\[thm:Serre\] If $\operatorname{End}_{{\overline{K}}}(E)={\mathbb{Z}}$, then $H_\infty$ is open in $\operatorname{GL}_2(\hat{{\mathbb{Z}}})$. Equivalently, the index of $H_N$ in $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ is bounded independently of $N$.

There is also a CM analogue of Theorem \[thm:Serre\], which is more easily stated by introducing the following definition:

\[def:MaximalRepresentation\] Let $E/K$ be an elliptic curve and $\ell$ be a prime number. We say that the image of the $\ell$-adic representation is 