show · modcurve.xfull all knowls · up · search:

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

Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2023-07-10 07:25:14
Referred to by:
History: (expand/hide all) Differences (show/hide)