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 | $4^{8}\cdot8^{2}$ | 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: | 8N0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}21&118\\40&29\end{bmatrix}$, $\begin{bmatrix}51&44\\56&63\end{bmatrix}$, $\begin{bmatrix}85&86\\52&85\end{bmatrix}$, $\begin{bmatrix}89&18\\84&77\end{bmatrix}$, $\begin{bmatrix}105&34\\4&47\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.48.0.k.2 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 $ | $=$ | $ 5 x^{2} + 4 x z + 6 y^{2} - 4 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.d.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-8.d.2.14 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.e.1.5 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.e.1.19 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.2.12 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-24.h.2.20 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.d.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.j.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bi.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bk.2.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bt.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.bv.2.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.ce.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-24.cf.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gx.2.11 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.gz.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hn.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.hp.2.13 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jj.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jl.2.16 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jz.2.16 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.kb.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-24.bg.1.32 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-24.z.2.28 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.u.1.15 | $120$ | $5$ | $5$ | $16$ |