Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot12^{2}\cdot16^{2}\cdot48^{2}$ | Cusp orbits | $2^{6}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 7$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48AP7 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}80&201\\47&154\end{bmatrix}$, $\begin{bmatrix}94&69\\109&230\end{bmatrix}$, $\begin{bmatrix}164&165\\119&202\end{bmatrix}$, $\begin{bmatrix}175&186\\46&227\end{bmatrix}$, $\begin{bmatrix}197&12\\204&125\end{bmatrix}$, $\begin{bmatrix}231&52\\238&237\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.192.7.yx.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $12$ |
| Cyclic 240-torsion field degree: | $384$ |
| Full 240-torsion field degree: | $1474560$ |
Rational points
This modular curve has no $\Q_p$ points for $p=47$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ |
| 240.96.0-240.dg.1.8 | $240$ | $4$ | $4$ | $0$ | $?$ |
| 240.192.3-24.gf.2.26 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.3-240.cho.1.29 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.3-240.cho.1.82 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.3-240.chs.1.77 | $240$ | $2$ | $2$ | $3$ | $?$ |
| 240.192.3-240.chs.1.115 | $240$ | $2$ | $2$ | $3$ | $?$ |