Invariants
| Level: | $264$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}38&73\\53&186\end{bmatrix}$, $\begin{bmatrix}105&172\\70&105\end{bmatrix}$, $\begin{bmatrix}120&55\\73&138\end{bmatrix}$, $\begin{bmatrix}165&172\\52&3\end{bmatrix}$, $\begin{bmatrix}230&231\\137&94\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 264.24.0.fi.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $48$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $20275200$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $6$ | $6$ | $0$ | $0$ |
| 88.6.0.c.1 | $88$ | $8$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.24.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 264.24.0-6.a.1.12 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.96.1-264.dg.1.18 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.gm.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.jw.1.24 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.jy.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.baa.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bac.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bag.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bai.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byv.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.byw.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzb.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzc.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzj.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzl.1.16 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzm.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1-264.bzo.1.24 | $264$ | $2$ | $2$ | $1$ |
| 264.144.1-264.bk.1.1 | $264$ | $3$ | $3$ | $1$ |