$X(N)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of the trivial subgroup of $\GL_2(\Z/N\Z)$. As a moduli space it parameterizes triples $(E,P,Q)$, where $E$ is an elliptic curve over $k$, and $P,Q \in E(k)$ form a basis for $E[N](\overline{k})$. There are other variants.

The canonical field of definition of $X(N)$ is $\Q(\zeta_N)$, which means that the database of modular curves $X_H/\Q$ only includes $X(N)$ for $N\le 2$.

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- Review status: beta
- Last edited by Andrew Sutherland on 2023-07-10 07:25:14

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**History:**(expand/hide all)

- 2023-07-10 07:25:14 by Andrew Sutherland
- 2023-07-09 08:59:20 by Andrew Sutherland
- 2023-07-09 08:52:45 by Andrew Sutherland
- 2022-11-06 16:41:59 by Ciaran Schembri

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