Image of $\ell$-adic Galois representation (reviewed)

Let $E/K$ be an elliptic curve over a number field, let $\ell$ be a prime, and let $H$ be the image of the $\ell$-adic Galois representation \[ \rho_{E,\ell}\colon G_\Q\to \Aut(E[\ell^{\infty}])\simeq \GL_2(\Z_\ell). \]

If $E$ does not have potential complex multiplication, then $H$ is an open subgroup of $\GL_2(\Z_\ell)$ that is equal to the inverse image of its projection to $\GL_2(\Z/\ell^n\Z)$, where $\ell^n$ is the level of $H$.

If $E$ has potential complex multiplication by an imaginary quadratic order $\mathcal O$, then $H$ is an open subgroup of a Cartan subgroup of $\GL_2(\Z_\ell)$ or its normalizer, depending on whether $K$ contains $\mathcal O$ or not. The particular Cartan subgroup depends on $\mathcal O$ and $\ell$, but in every case $H $ is determined by its projection to $\GL_2(\Z/\ell^n\Z)$, where $n$ is $4,3,1,1,\ldots$ for $\ell=2,3,5,7,\ldots$.

The subgroup $H$ can be identified (up to conjugacy) by giving the label of its projection to $\GL_2(\Z/\ell^n\Z)$ as defined by by Rouse, Sutherland, and Zureick-Brown in [arXiv:2106.11141].