Invariants
| Level: | $240$ | $\SL_2$-level: | $80$ | Newform level: | $400$ | ||
| Index: | $480$ | $\PSL_2$-index: | $240$ | ||||
| Genus: | $16 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $4$ are rational) | Cusp widths | $10^{4}\cdot20^{2}\cdot40^{4}$ | Cusp orbits | $1^{4}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 16$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40B16 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}15&112\\184&177\end{bmatrix}$, $\begin{bmatrix}19&124\\96&55\end{bmatrix}$, $\begin{bmatrix}91&74\\208&135\end{bmatrix}$, $\begin{bmatrix}93&10\\208&17\end{bmatrix}$, $\begin{bmatrix}111&2\\184&87\end{bmatrix}$, $\begin{bmatrix}151&120\\120&41\end{bmatrix}$, $\begin{bmatrix}207&50\\64&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.240.16.bp.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| Full 240-torsion field degree: | $1179648$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $96$ | $48$ | $0$ | $0$ |
| 48.96.0-8.l.1.4 | $48$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.96.0-8.l.1.4 | $48$ | $5$ | $5$ | $0$ | $0$ |
| 240.240.8-40.v.1.11 | $240$ | $2$ | $2$ | $8$ | $?$ |
| 240.240.8-40.v.1.14 | $240$ | $2$ | $2$ | $8$ | $?$ |
| 240.240.8-40.dd.2.3 | $240$ | $2$ | $2$ | $8$ | $?$ |
| 240.240.8-40.dd.2.7 | $240$ | $2$ | $2$ | $8$ | $?$ |
| 240.240.8-40.dd.2.10 | $240$ | $2$ | $2$ | $8$ | $?$ |
| 240.240.8-40.dd.2.14 | $240$ | $2$ | $2$ | $8$ | $?$ |