Invariants
| Level: | $240$ | $\SL_2$-level: | $48$ | Newform level: | $1$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot48^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 9$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 48B9 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}28&87\\45&34\end{bmatrix}$, $\begin{bmatrix}28&179\\41&170\end{bmatrix}$, $\begin{bmatrix}130&93\\171&208\end{bmatrix}$, $\begin{bmatrix}167&170\\238&179\end{bmatrix}$, $\begin{bmatrix}184&211\\149&58\end{bmatrix}$, $\begin{bmatrix}221&76\\178&67\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.144.9.zn.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $1966080$ |
Rational points
This modular curve has no $\Q_p$ points for $p=127$, 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
| 80.96.1-80.cf.1.15 | $80$ | $3$ | $3$ | $1$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.144.4-48.bf.2.7 | $48$ | $2$ | $2$ | $4$ | $0$ |
| 80.96.1-80.cf.1.15 | $80$ | $3$ | $3$ | $1$ | $?$ |
| 120.144.4-120.ot.1.19 | $120$ | $2$ | $2$ | $4$ | $?$ |
| 240.144.4-48.bf.2.12 | $240$ | $2$ | $2$ | $4$ | $?$ |
| 240.144.4-120.ot.1.49 | $240$ | $2$ | $2$ | $4$ | $?$ |
| 240.144.5-240.b.1.114 | $240$ | $2$ | $2$ | $5$ | $?$ |
| 240.144.5-240.b.1.123 | $240$ | $2$ | $2$ | $5$ | $?$ |