of CM elliptic curves, we know that there are finitely many values $j_1,\ldots,j_k \in \overline{{\mathbb{Q}}}$ such that for every CM elliptic curve $E/K$ we have $j(E) \in \{j_1,\ldots,j_k\}$. For each such $j_i$, fix an elliptic curve $E_i/K$ with $j(E_i)=j_i$. To every $E_i/K$ corresponds a minimal parameter of maximal growth for the $\ell$-adic representation that we call $m_i$. Let $n_\ell=\max\{n, m_i+2 \bigm\vert i =1,\ldots,k\}$: we claim that this value of $n_\ell$ satisfies the conclusion of the Corollary. Indeed, let $E/K$ be any elliptic curve. If $E$ does not have CM, the minimal parameter of maximal growth for its $\ell$-adic representation is at most $n {\leqslant}n_\ell$. If $E$ has CM, then there exists $i$ such that $j(E)=j_i=j(E_i)$, so $E$ is a twist of $E_i$. By Lemma \[lemma:Untwisting\] the curves $E$ and $E_i$ become isomorphic over an extension $F/K$ of degree dividing $12$, so if $m$ (resp. $\tilde{m}$, resp. $\tilde{m}_i$) denotes the minimal parameter of maximal growth for $E/K$ (resp. for $E/F$, resp. for $E_i/F$) we have $$m {\leqslant}\tilde{m} = \tilde{m}_i {\leqslant}m_i + 2 {\leqslant}n_\ell,$$ where the equality follows from the fact that $E$ and $E_i$ are isomorphic over $F$, while the inequality $\tilde{m}_i {\leqslant}m_i + 2$ follows from Lemma \[lem-param-tilde-bound\] and the 