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 | $1^{4}\cdot2^{2}\cdot4^{2}\cdot16^{2}$ | 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: | 16H0 |
Level structure
| $\GL_2(\Z/240\Z)$-generators: | $\begin{bmatrix}7&212\\168&211\end{bmatrix}$, $\begin{bmatrix}134&37\\1&226\end{bmatrix}$, $\begin{bmatrix}197&10\\126&89\end{bmatrix}$, $\begin{bmatrix}225&178\\116&231\end{bmatrix}$, $\begin{bmatrix}226&101\\63&224\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.cs.1 for the level structure with $-I$) |
| Cyclic 240-isogeny field degree: | $48$ |
| Cyclic 240-torsion field degree: | $3072$ |
| 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 |
|---|---|---|---|---|---|
| 48.48.0-24.by.1.3 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-80.n.1.22 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.by.1.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-80.n.1.13 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.o.1.34 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.o.1.55 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.be.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.cr.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ep.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.fj.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jh.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jn.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kl.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kv.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lv.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.me.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nc.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nj.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oj.2.10 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ol.2.14 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.pj.2.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.px.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.to.1.31 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.yj.1.47 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.du.2.23 | $240$ | $5$ | $5$ | $16$ |