prime. The following hold:

(1) The image $H_p$ of the modulo-$p$ representation attached to $E$ contains $\{\lambda\operatorname{Id}\mid \lambda\in{\mathbb{F}}_p^\times\}$.

(2) Suppose $H_p\neq \operatorname{GL}_2({\mathbb{F}}_p)$ and let $g_p\in \operatorname{GL}_2({\mathbb{F}}_p)$ be an element that normalises $H_p$. Then there is $h\in \operatorname{GL}_2({\mathbb{F}}_p)$ such that $h^{-1}g_ph\in N_{\operatorname{ns}}(p)$ and $h^{-1}H_ph\subseteq N_{\operatorname{ns}}(p)$.

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(1) We apply Theorem \[thm:Zywyna\]. If $H_p$ is either $\operatorname{GL}_2({\mathbb{F}}_p)$ or conjugate to $N_{\operatorname{ns}}(p)$, the conclusion follows trivially, since $C_{\operatorname{ns}}(p)$ contains all scalars. In case (iii) of Theorem \[thm:Zywyna\], $H_p$ contains the cubes of the scalars, hence all scalars since $p\equiv 2\pmod 3$.

(2) We only have to consider cases (ii) and (iii) of Theorem \[thm:Zywyna\]. Up to conjugation, we may assume that $H_p\subseteq N_{\operatorname{ns}}(p)$ and the claim becomes $g_p\in N_{\operatorname{ns}}(p)$.

    In case (ii) it suffices to check that the normaliser of $N_{\operatorname{ns}}(p)$ is $N_{\operatorname{ns}}(p)$ itself. This holds because $C_{\operatorname{ns}}(p)$, being the only cyclic subgroup of index $2$ of $N_{\operatorname{ns}}(p)$, is characteristic in $N_{\operatorname{ns}}(p)$; hence any element that normalises $N_{\operatorname{ns}}(p)$ normalises $C_{\operatorname{ns}}(p)$ as well, so it must be in $N_{\operatorname{ns}}(p)$. In case (iii), one similarly sees that $\{a^3\mid a\in C_{\operatorname{ns}}(p)\}$ is characteristic in $D(p)$ and that its normaliser is $N_{\operatorname{ns}}(p)$, and the conclusion follows as above.

\[lem:LiftHomothety\] Let $\ell$ be a prime number and 