Invariants
| Level: | $240$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| 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: | 16G0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}41&12\\172&155\end{bmatrix}$, $\begin{bmatrix}41&144\\207&235\end{bmatrix}$, $\begin{bmatrix}87&208\\62&31\end{bmatrix}$, $\begin{bmatrix}153&4\\154&59\end{bmatrix}$, $\begin{bmatrix}169&20\\79&129\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.bl.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $1536$ |
| Full 240-torsion field degree: | $5898240$ |
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 |
|---|---|---|---|---|---|
| 48.48.0-48.f.2.9 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-40.bl.1.7 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-40.bl.1.11 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.f.2.21 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.n.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.n.1.51 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.192.1-240.hv.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.hw.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.id.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ie.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oh.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oi.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ox.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oy.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wx.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wy.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xn.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xo.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbp.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbq.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bbx.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bby.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.fe.1.7 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.sh.2.5 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.cn.1.10 | $240$ | $5$ | $5$ | $16$ |