elliptic curves:

\[lemma:Untwisting\] Let $E_1, E_2$ be elliptic curves over $K$ such that $(E_1)_{\overline{{\mathbb{Q}}}}$ is isomorphic to $(E_2)_{\overline{{\mathbb{Q}}}}$. There is an extension $F$ of $K$, of degree dividing $12$, such that $E_1$ and $E_2$ become isomorphic over $F$.

Fixing a $\overline{{\mathbb{Q}}}$-isomorphism between $E_1$ and $E_2$ allows us to attach to $E_2$ a class in $H^1\left( \operatorname{Gal}(\overline{K} \mid K), \operatorname{Aut}(E_1) \right)$. Since $H^1\left( \operatorname{Gal}(\overline{K} \mid K), \operatorname{Aut}(E_1) \right) \cong K^\times/K^{\times n}$ for some $n \in \{2, 4, 6\}$ (see [@SilvermanEC Proposition X.5.4]), the class of $E_2$ corresponds to the class of a certain $[\alpha] \in K^\times/K^{\times n}$. Letting $F=K(\sqrt[n]{\alpha})$, whose degree over $K$ divides $12$, it is clear that $[\alpha] \in F^\times/F^{\times n}$ is the trivial class, so the same is true for $[E_2] \in H^1(\operatorname{Gal}(\overline{F} \mid F), \operatorname{Aut}(E_1))$, which means that $E_2$ is isomorphic to $E_1$ over $F$ as desired.

\[cor:UniversalBoundGrowthParameter\] Let $K$ be a number field and $\ell$ be a prime number. There exists an integer $n_\ell$ with the following property: for every elliptic curve $E/K$, the minimal parameter of maximal growth for the $\ell$-adic representation attached to $E$ is at most $n_\ell$.

Let $n$ be the integer whose existence is guaranteed by Theorem \[thm:Arai\]. By the general theory 