Invariants
| Level: | $264$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}19&112\\130&183\end{bmatrix}$, $\begin{bmatrix}37&204\\76&89\end{bmatrix}$, $\begin{bmatrix}49&8\\58&79\end{bmatrix}$, $\begin{bmatrix}75&188\\158&191\end{bmatrix}$, $\begin{bmatrix}249&136\\38&255\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 264.48.0.bu.2 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $96$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $10137600$ |
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 |
|---|---|---|---|---|---|
| 8.48.0-8.e.2.13 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 264.48.0-8.e.2.11 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.u.1.33 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.u.1.48 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.x.1.7 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.x.1.35 | $264$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.192.1-264.bz.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ce.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.cy.2.9 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.db.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ea.2.14 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.eb.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.eg.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.eh.2.7 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.gq.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.gr.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.gs.2.8 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.gt.2.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.hg.2.5 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.hh.1.11 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.hi.2.11 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.hj.1.9 | $264$ | $2$ | $2$ | $1$ |
| 264.288.8-264.mc.1.52 | $264$ | $3$ | $3$ | $8$ |
| 264.384.7-264.gp.1.50 | $264$ | $4$ | $4$ | $7$ |