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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 $X_H$.
Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2022-11-06 20:28:07
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