Image of the adelic Galois representation (awaiting review)

The image of the adelic Galois representation associate to an elliptic curve $E$ over a number field $K$ that does not have potential complex multiplication is an open subgroup $H$ of $\GL(2,\widehat\Z)$. The subgroup $H$ has the following invariants:

• The level of $H$ is the least positive integer $N$ such that $H$ is the full inverse image of its projection to $\GL(2,\Z/N\Z)$.
• The index of $H$ is the positive integer $[\GL(2,\Z/N\Z):H]$.
• The genus of $H$ is the genus of the corresponding modular curve $X_H$.