Invariants
| Level: | $264$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}109&144\\25&161\end{bmatrix}$, $\begin{bmatrix}181&160\\243&55\end{bmatrix}$, $\begin{bmatrix}191&124\\102&25\end{bmatrix}$, $\begin{bmatrix}215&240\\17&205\end{bmatrix}$, $\begin{bmatrix}249&56\\119&113\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 132.12.0.h.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $96$ |
| Cyclic 264-torsion field degree: | $7680$ |
| Full 264-torsion field degree: | $40550400$ |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.12.0-4.c.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 88.12.0-4.c.1.3 | $88$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.72.2-132.x.1.17 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-132.p.1.23 | $264$ | $4$ | $4$ | $1$ |
| 264.288.9-132.p.1.28 | $264$ | $12$ | $12$ | $9$ |
| 264.48.0-264.da.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.da.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.db.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.db.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.de.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.de.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.df.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.df.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.du.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.du.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dv.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dv.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dy.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dy.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dz.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dz.1.9 | $264$ | $2$ | $2$ | $0$ |