$X(N)$ (awaiting review)

$X(N)$ 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 $N\le 2$.