Invariants
| Level: | $264$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
Level structure
| $\GL_2(\Z/264\Z)$-generators: | $\begin{bmatrix}149&190\\153&169\end{bmatrix}$, $\begin{bmatrix}151&90\\30&175\end{bmatrix}$, $\begin{bmatrix}205&50\\138&179\end{bmatrix}$, $\begin{bmatrix}233&66\\81&161\end{bmatrix}$, $\begin{bmatrix}233&210\\198&71\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 264.288.5-264.bpr.1.1, 264.288.5-264.bpr.1.2, 264.288.5-264.bpr.1.3, 264.288.5-264.bpr.1.4, 264.288.5-264.bpr.1.5, 264.288.5-264.bpr.1.6, 264.288.5-264.bpr.1.7, 264.288.5-264.bpr.1.8 |
| Cyclic 264-isogeny field degree: | $48$ |
| Cyclic 264-torsion field degree: | $3840$ |
| Full 264-torsion field degree: | $6758400$ |
Rational points
This modular curve has no $\Q_p$ points for $p=31$, and therefore no 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $12$ | $12$ | $0$ | $0$ |
| 88.12.0.bl.1 | $88$ | $12$ | $12$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.3.ug.1 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 132.72.1.h.1 | $132$ | $2$ | $2$ | $1$ | $?$ |
| 264.48.1.bzp.1 | $264$ | $3$ | $3$ | $1$ | $?$ |
| 264.72.1.bn.1 | $264$ | $2$ | $2$ | $1$ | $?$ |
| 264.72.1.id.1 | $264$ | $2$ | $2$ | $1$ | $?$ |
| 264.72.3.cvg.1 | $264$ | $2$ | $2$ | $3$ | $?$ |
| 264.72.3.cvt.1 | $264$ | $2$ | $2$ | $3$ | $?$ |
| 264.72.3.ele.1 | $264$ | $2$ | $2$ | $3$ | $?$ |