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Let $M$ and $N$ denote positive integers and $\ell$ a prime. Each modular curve below "parametrizes" elliptic curves with a particular type of level structure. Standard modular curves include:

$X_0(N)$, $X_1(N)$, $X_{\pm 1}(N)$, $X_1(M,MN)$, $X_{\pm 1}(M,MN)$, $X(N)$, $X_{\mathrm sp}(N)$, $X_{\mathrm sp}^+(N)$, $X_{\mathrm ns}(N)$, $X_{\mathrm ns}^+(N)$, $X_{S_4}(\ell)$, $X_{\mathrm sym}(N)$

The curves above can all be formally defined as special cases of the modular curve $X_H$ determined by an open subgroup $H \le \GL_2(\widehat{\Z})$ of level $N$, $MN$, or $\ell$.

Knowl status:
  • Review status: beta
  • Last edited by Yongyuan Huang on 2023-08-01 13:10:16
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