Let $E / \mathbb{Q}$ be the elliptic curve given by $y^{2} = x^{3} - x$. I would like to know explicitly what the field of all $2$-power torsion looks like, as well as the image in $\mathrm{GL}(T_{2}(E))$ of the $2$-adic Galois representation (and I would be interested in analogous descriptions for $\ell$-adic Galois representations with $\ell \neq 2$ as well). I'm sure this could be computed using Sage or Magma, but I thought I'd ask to see if it's already written down somewhere.

Here's what I do know:

1) Since $E$ has complex multiplication with $\mathrm{End}_{\mathbb{Q}(i)}(E) \cong \mathbb{Z}[i]$, and $j(E) \in \mathbb{Q}$, the dyadic torsion field must be an abelian extension of $\mathbb{Q}(i)$ ramified only above the prime $(1 + i)$. In fact, it must be the maximal such abelian extension.

2) The dyadic torsion field must contain all $2$-power roots of unity.

3) The division field $\mathbb{Q}(E[8])$ is $\mathbb{Q}(\zeta_{16}, 2^{1/4})$, and the image of $\mathrm{Gal}(\bar{\mathbb{Q}} / \mathbb{Q}(\zeta_{8}))$ modulo $8$ is generated by the matrices $$\left[ {\begin{array}{cc} 1 & 4 \\ 4 & 1 \end{array} } \right], \left[ {\begin{array}{cc} 5 & 0 \\ 0 & 5 \end{array} } \right] \in \mathrm{SL}_{2}(\mathbb{Z} / 8\mathbb{Z})$$ with respect to a suitable basis.

[EDIT: I corrected a mistake I noticed in my claim (3) above.]