  For a non-CM elliptic curve $E$ over the rationals, the Galois action on its
torsion points can be expressed in terms of a Galois representation $\rho_E : G
\to GL_2(\hat{\mathbb{Z}})$, where $G$ is the absolute Galois group of the
rationals. A well-known theorem of Serre says that the image of $\rho_E$ is
open and hence has finite index in $GL_2(\hat{\mathbb{Z}})$. We will study what
indices are possible assuming that we are willing to exclude a finite number of
possible $j$-invariants from consideration. For example, we will show that
there is a finite set $J$ of rational numbers such that if $E/\mathbb{Q}$ is a
non-CM elliptic curve with $j$-invariant not in $ J$ and with surjective mod
$\ell$ representations for all $\ell >37$ (which conjecturally always holds),
then the index $[GL_2(\hat{\mathbb{Z}}) : \rho_E(G)]$ lies in the set \[
  I:= \left\{\begin{array}{c}2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 32, 36, 40,
48, 54, 60, 72, 84, 96, 108, 112,120, 144, \\192, 220, 240, 288, 336, 360, 384,
504, 576, 768, 864, 1152, 1200, 1296, 1536 \end{array}\right\}. \] Moreover,
$I$ is the minimal set with this property.
