  For $E/K$ an elliptic curve without complex multiplication we bound the index
of the image of $\operatorname{Gal}(\bar{K}/K)$ in
$\operatorname{GL}_2(\hat{\mathbb{Z}})$, the representation being given by the
action on the Tate modules of $E$ at the various primes. The bound is effective
and only depends on $[K:\mathbb{Q}]$ and on the stable Faltings height of $E$.
We also prove a result relating the structure of subgroups of
$\operatorname{GL}_2(\mathbb{Z}_\ell)$ to certain Lie algebras naturally
attached to them.
