Invariants
| Level: | $240$ | $\SL_2$-level: | $80$ | Newform level: | $1$ | ||
| Index: | $240$ | $\PSL_2$-index: | $120$ | ||||
| Genus: | $8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $5^{2}\cdot10\cdot20\cdot40^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | $3 \le \gamma \le 8$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 40C8 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}98&21\\127&32\end{bmatrix}$, $\begin{bmatrix}106&129\\105&202\end{bmatrix}$, $\begin{bmatrix}131&22\\66&119\end{bmatrix}$, $\begin{bmatrix}147&62\\224&17\end{bmatrix}$, $\begin{bmatrix}192&161\\77&4\end{bmatrix}$, $\begin{bmatrix}233&40\\180&173\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.120.8.gg.2 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $2359296$ |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|
| $X_{S_4}(5)$ | $5$ | $48$ | $24$ | $0$ | $0$ |
| 48.48.0-24.by.2.7 | $48$ | $5$ | $5$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 48.48.0-24.by.2.7 | $48$ | $5$ | $5$ | $0$ | $0$ |
| 80.120.4-40.bl.1.5 | $80$ | $2$ | $2$ | $4$ | $?$ |
| 240.120.4-40.bl.1.3 | $240$ | $2$ | $2$ | $4$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.