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 | $2\cdot4^{2}$ | ||
| 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: | $\begin{bmatrix}1&8\\16&229\end{bmatrix}$, $\begin{bmatrix}59&12\\77&113\end{bmatrix}$, $\begin{bmatrix}123&188\\50&19\end{bmatrix}$, $\begin{bmatrix}125&4\\168&77\end{bmatrix}$, $\begin{bmatrix}139&48\\18&215\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.o.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $96$ |
| Cyclic 240-torsion field degree: | $6144$ |
| 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 |
|---|---|---|---|---|---|
| 40.48.0-40.w.1.7 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 240.48.0-40.w.1.4 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.192.3-240.wn.1.1 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.wo.1.8 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.wp.1.8 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.wq.1.5 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.wz.1.2 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.xa.1.7 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.xb.1.7 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.xc.1.6 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zb.1.15 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zc.1.10 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zd.1.12 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.ze.1.15 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zn.1.16 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zo.1.9 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zp.1.11 | $240$ | $2$ | $2$ | $3$ |
| 240.192.3-240.zq.1.16 | $240$ | $2$ | $2$ | $3$ |
| 240.288.8-240.cf.1.64 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.pa.1.47 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.ba.1.28 | $240$ | $5$ | $5$ | $16$ |