some results on the growth of the torsion extensions from [@2016arXiv161202847L §2.3].

\[prop-max-growth\] Let $\ell$ be a prime number. Let $\delta=2$ if $E$ has complex multiplication and $\delta=4$ otherwise. There exists a positive integer $n_\ell$ such that $$\begin{aligned}
\#H_{\ell^{n+1}}/\#H_{\ell^n}=\ell^\delta \qquad \text{for every }n{\geqslant}n_\ell.\end{aligned}$$

This follows from Theorem \[thm:Serre\] in the non-CM case and from classical results in the CM case. See also [@2016arXiv161202847L Lemma 10 and Remark 13] for a more general result.

We call an integer $n_\ell$ as in Proposition \[prop-max-growth\] a *parameter of maximal growth for the $\ell$-adic torsion representation*. We say that it is *minimal* if $n_\ell-1$ is not a parameter of maximal growth; when $\ell=2$, we require that the minimal parameter be at least 2.

The assumption $n_\ell {\geqslant}2$ when $\ell=2$ will be needed to apply [@2016arXiv161202847L Theorem 12].

\[rmk:nlIsEffective\] Given an explicit elliptic curve $E/K$ and a prime $\ell$, the problem of determining the optimal value of $n_\ell$ can be solved effectively (see [@2016arXiv161202847L Remark 13]). However, computing $n_\ell$ can be challenging in practice, because the naïve algorithm requires the determination of the Galois groups of the splitting fields of several large-degree polynomials. The situation is usually better for smaller primes $\ell$, and especially for 