fact that $v_\ell([F:K]) {\leqslant}v_\ell(12) {\leqslant}2$ for every prime $\ell$.

Possible images of $\bmod\,\ell$ representations {#subsec:PossibleImagesModl}
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We recall several results concerning the images of the $\bmod\,\ell$ representations attached to elliptic curves over ${\mathbb{Q}}$.

We begin with a famous Theorem of Mazur. Let $\mathcal{T}_0:={\left\{ p\text{ prime }\mid p{\leqslant}17 \right\}}\cup\{37\}$.

\[thm:Mazur\] Let $E/{\mathbb{Q}}$ be an elliptic curve and assume that $E$ has a $\mathbb{Q}$-rational subgroup of order $p$. Then $p\in \mathcal T_0\cup {\left\{ 19,43,67,163 \right\}}$. If $E$ does not have CM over $\overline {\mathbb{Q}}$, then $p\in\mathcal{T}_0$.

\[thm:Zywyna\] Let $E/{\mathbb{Q}}$ be a non-CM elliptic curve and $p\not \in \mathcal{T}_0$ be a prime. Let $C_{\operatorname{ns}}(p)$ be the subgroup of $\operatorname{GL}_2({\mathbb{F}}_p)$ consisting of all matrices of the form $\begin{pmatrix}
a & b\epsilon\\
b & a
\end{pmatrix}$ with $(a,b)\in{\mathbb{F}}_p^2\setminus \{(0,0)\}$ and $\epsilon$ a fixed element of ${\mathbb{F}}_p^\times\setminus {\mathbb{F}}_p^{\times 2}$. Then $H_p$ is conjugate to one of the following:

(i) $\operatorname{GL}_2({\mathbb{F}}_p)$;

(ii) the normaliser $N_{\operatorname{ns}}(p)$ of $C_{\operatorname{ns}}(p)$;

(iii) the index $3$ subgroup $$D(p):={\left\{ a^3\mid a\in C_{\operatorname{ns}}(p) \right\}}\cup {\left\{ \begin{pmatrix}
    1 & 0\\
    0 & -1
    \end{pmatrix}\cdot a^3\mid a\in C_{\operatorname{ns}}(p) \right\}}$$ of $N_{\operatorname{ns}}(p)$.

Moreover, the last case can only occur if $p\equiv 2\pmod 3$.

\[cor:ContainsScalarsAndConjugation\] Let $E/{\mathbb{Q}}$ be a non-CM elliptic curve and $p\not \in \mathcal{T}_0$ be a 