Invariants
| Level: | $264$ | $\SL_2$-level: | $4$ | ||||
| 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: | $\begin{bmatrix}47&158\\252&163\end{bmatrix}$, $\begin{bmatrix}55&134\\124&237\end{bmatrix}$, $\begin{bmatrix}115&192\\82&205\end{bmatrix}$, $\begin{bmatrix}121&222\\212&65\end{bmatrix}$, $\begin{bmatrix}145&32\\134&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 132.12.0.a.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $192$ |
| Cyclic 264-torsion field degree: | $15360$ |
| 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 |
|---|---|---|---|---|---|
| 8.12.0-2.a.1.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 264.12.0-2.a.1.2 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.48.0-132.a.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.b.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.b.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.c.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.c.1.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.d.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.d.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.f.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.f.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.g.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.g.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.j.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.j.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.s.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.s.1.9 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-132.c.1.4 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-132.c.1.9 | $264$ | $4$ | $4$ | $1$ |
| 264.288.9-132.c.1.13 | $264$ | $12$ | $12$ | $9$ |