Invariants
| Level: | $240$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}0&49\\109&14\end{bmatrix}$, $\begin{bmatrix}18&55\\85&108\end{bmatrix}$, $\begin{bmatrix}36&31\\233&186\end{bmatrix}$, $\begin{bmatrix}167&18\\102&115\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.24.0.l.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $192$ |
| Cyclic 240-torsion field degree: | $12288$ |
| Full 240-torsion field degree: | $11796480$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-40.p.1.4 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 240.24.0-40.p.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.144.4-240.bn.1.15 | $240$ | $3$ | $3$ | $4$ |
| 240.192.3-240.chg.1.28 | $240$ | $4$ | $4$ | $3$ |
| 240.240.8-240.p.1.8 | $240$ | $5$ | $5$ | $8$ |
| 240.288.7-240.of.1.26 | $240$ | $6$ | $6$ | $7$ |
| 240.480.15-240.bb.1.14 | $240$ | $10$ | $10$ | $15$ |