Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| 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 | $2^{3}\cdot4$ | ||
| 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: | 8O0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}3&100\\74&109\end{bmatrix}$, $\begin{bmatrix}31&116\\26&75\end{bmatrix}$, $\begin{bmatrix}33&100\\58&97\end{bmatrix}$, $\begin{bmatrix}51&64\\118&89\end{bmatrix}$, $\begin{bmatrix}71&112\\2&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.48.0.ct.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $768$ |
| Full 120-torsion field degree: | $368640$ |
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 |
|---|---|---|---|---|---|
| 24.48.0-24.m.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.i.2.17 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-40.i.2.30 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.m.1.8 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.t.2.41 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.t.2.51 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.192.1-120.x.2.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.bc.2.16 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.cj.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.cm.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.fi.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.fj.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.fm.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.fn.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ng.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nh.2.15 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ns.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nt.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nw.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nx.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.oi.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.oj.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-120.ot.2.29 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.jg.1.51 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.do.2.11 | $120$ | $5$ | $5$ | $16$ |