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: | $\begin{bmatrix}39&172\\197&95\end{bmatrix}$, $\begin{bmatrix}113&64\\139&175\end{bmatrix}$, $\begin{bmatrix}113&172\\205&201\end{bmatrix}$, $\begin{bmatrix}127&144\\91&133\end{bmatrix}$, $\begin{bmatrix}155&152\\28&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 240.48.0.bx.2 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 |
|---|---|---|---|---|---|
| 48.48.0-48.e.1.29 | $48$ | $2$ | $2$ | $0$ | $0$ |
| 80.48.0-80.m.2.27 | $80$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.df.1.19 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-48.e.1.22 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-80.m.2.13 | $240$ | $2$ | $2$ | $0$ | $?$ |
| 240.48.0-120.df.1.12 | $240$ | $2$ | $2$ | $0$ | $?$ |
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.jf.2.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jg.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jv.2.19 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.jw.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sv.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.sw.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.tt.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.tu.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.yz.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.za.2.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zx.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.zy.2.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bcz.1.21 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bda.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdp.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.bdq.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.288.8-240.gn.2.39 | $240$ | $3$ | $3$ | $8$ |
| 240.384.7-240.tk.1.29 | $240$ | $4$ | $4$ | $7$ |
| 240.480.16-240.cz.2.30 | $240$ | $5$ | $5$ | $16$ |