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: | $\begin{bmatrix}5&12\\218&151\end{bmatrix}$, $\begin{bmatrix}163&220\\130&101\end{bmatrix}$, $\begin{bmatrix}211&4\\68&43\end{bmatrix}$, $\begin{bmatrix}220&203\\121&70\end{bmatrix}$, $\begin{bmatrix}221&24\\216&157\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.dk.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
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.2.1 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-40.ca.2.16 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-40.ca.2.12 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.f.2.5 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.p.1.19 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-240.p.1.33 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.bn.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.da.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.es.1.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ft.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.iz.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jy.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.kg.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ld.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.lj.2.3 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mn.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.mv.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nn.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.nz.2.5 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oy.1.9 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.pg.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.qd.1.1 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.uw.1.3 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.zb.2.1 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.em.2.1 | $240$ | $5$ | $5$ | $16$ |