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 $X_H$.

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- Review status: beta
- Last edited by Andrew Sutherland on 2022-11-06 20:28:07

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