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 | $2^{5}$ | ||
| 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
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}31&8\\58&15\end{bmatrix}$, $\begin{bmatrix}55&56\\118&85\end{bmatrix}$, $\begin{bmatrix}93&100\\86&27\end{bmatrix}$, $\begin{bmatrix}115&68\\82&55\end{bmatrix}$, $\begin{bmatrix}117&20\\40&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.v.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $368640$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ x^{2} + x z - 6 y^{2} + 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 |
|---|---|---|---|---|---|
| 40.48.0-8.e.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.e.1.15 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.1.26 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.1.31 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.m.1.3 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.m.1.18 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.192.1-24.h.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.i.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.x.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.y.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bk.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bl.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bo.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bp.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-24.ep.2.30 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-24.cw.2.17 | $120$ | $4$ | $4$ | $7$ |
| 120.192.1-120.nk.2.13 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nl.2.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.no.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.np.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.oa.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ob.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.oe.2.9 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.of.2.15 | $120$ | $2$ | $2$ | $1$ |
| 120.480.16-120.dp.2.14 | $120$ | $5$ | $5$ | $16$ |