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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{5}$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}5&88\\151&231\end{bmatrix}$, $\begin{bmatrix}29&128\\167&57\end{bmatrix}$, $\begin{bmatrix}105&88\\211&89\end{bmatrix}$, $\begin{bmatrix}175&72\\112&7\end{bmatrix}$, $\begin{bmatrix}199&8\\137&167\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.bj.2 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
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 40 x^{2} - y^{2} - 5 z^{2} $ |
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 |
|---|---|---|---|---|---|
| 48.48.0-8.bb.1.2 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 240.48.0-8.bb.1.6 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-40.bj.1.2 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-40.bj.1.7 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-40.cb.2.1 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-40.cb.2.11 | $240$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 240.192.1-80.cf.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.ch.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.cn.2.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.cp.1.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.dn.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.dp.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.dv.1.2 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-80.dx.2.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ny.2.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oa.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oo.2.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oq.2.8 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wo.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wq.2.4 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xe.2.6 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xg.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-120.rj.1.31 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-120.li.1.23 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-40.bz.1.4 | $240$ | $5$ | $5$ | $16$ |