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 | $1^{2}\cdot2\cdot8$ | 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: | 8C0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}28&247\\73&50\end{bmatrix}$, $\begin{bmatrix}61&18\\52&247\end{bmatrix}$, $\begin{bmatrix}84&41\\137&252\end{bmatrix}$, $\begin{bmatrix}87&68\\256&235\end{bmatrix}$, $\begin{bmatrix}215&220\\216&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 264.12.0.z.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 |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 264.12.0-4.c.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 |
|---|---|---|---|---|
| 264.48.0-264.y.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ba.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bg.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bh.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bu.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bx.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.bz.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ca.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cl.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cm.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.co.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cr.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.db.1.24 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dc.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dq.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dt.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.72.2-264.cx.1.62 | $264$ | $3$ | $3$ | $2$ |
| 264.96.1-264.zv.1.63 | $264$ | $4$ | $4$ | $1$ |
| 264.288.9-264.ijz.1.64 | $264$ | $12$ | $12$ | $9$ |