Invariants
| Level: | $264$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12K3 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}67&160\\150&239\end{bmatrix}$, $\begin{bmatrix}101&192\\232&157\end{bmatrix}$, $\begin{bmatrix}133&154\\18&17\end{bmatrix}$, $\begin{bmatrix}153&172\\218&29\end{bmatrix}$, $\begin{bmatrix}181&244\\84&167\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 264.96.3.cr.1 for the level structure with $-I$) |
| Cyclic 264-isogeny field degree: | $48$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $5068800$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $24$ | $24$ | $0$ | $0$ |
| 88.24.0.b.1 | $88$ | $8$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-24.bw.1.10 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 132.96.1-132.a.1.17 | $132$ | $2$ | $2$ | $1$ | $?$ |
| 264.96.1-132.a.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ |
| 264.96.1-24.bw.1.13 | $264$ | $2$ | $2$ | $1$ | $?$ |
| 264.96.1-264.dg.1.18 | $264$ | $2$ | $2$ | $1$ | $?$ |
| 264.96.1-264.dg.1.35 | $264$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 264.384.5-264.ip.1.10 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ip.2.13 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ip.3.10 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ip.4.13 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ir.1.6 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ir.2.8 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ir.3.16 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ir.4.15 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ob.1.9 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ob.2.14 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ob.3.9 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.ob.4.14 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.od.1.8 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.od.2.7 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.od.3.12 | $264$ | $2$ | $2$ | $5$ |
| 264.384.5-264.od.4.16 | $264$ | $2$ | $2$ | $5$ |