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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}33&190\\248&11\end{bmatrix}$, $\begin{bmatrix}57&206\\128&187\end{bmatrix}$, $\begin{bmatrix}77&124\\28&63\end{bmatrix}$, $\begin{bmatrix}79&126\\44&109\end{bmatrix}$, $\begin{bmatrix}107&54\\204&109\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 264.48.0.bm.2 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $96$ |
| Cyclic 264-torsion field degree: | $7680$ |
| 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.1.15 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 264.48.0-8.e.1.4 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.e.1.14 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.e.1.39 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.u.2.32 | $264$ | $2$ | $2$ | $0$ | $?$ |
| 264.48.0-264.u.2.52 | $264$ | $2$ | $2$ | $0$ | $?$ |
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.ce.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.db.2.5 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.eq.1.2 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ew.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.in.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ip.1.14 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ki.2.15 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.kk.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.mj.2.5 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.ml.2.13 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.oe.1.4 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.og.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.pj.1.14 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.pp.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.qc.1.7 | $264$ | $2$ | $2$ | $1$ |
| 264.192.1-264.qf.1.8 | $264$ | $2$ | $2$ | $1$ |
| 264.288.8-264.dz.2.10 | $264$ | $3$ | $3$ | $8$ |
| 264.384.7-264.dn.2.14 | $264$ | $4$ | $4$ | $7$ |