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: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8N0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}51&100\\8&71\end{bmatrix}$, $\begin{bmatrix}67&38\\68&11\end{bmatrix}$, $\begin{bmatrix}71&110\\4&11\end{bmatrix}$, $\begin{bmatrix}83&46\\80&29\end{bmatrix}$, $\begin{bmatrix}95&26\\28&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.48.0.be.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
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.48.0-8.e.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-60.c.1.21 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-60.c.1.23 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.t.2.12 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.t.2.38 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.be.2.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.da.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.dy.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.eg.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.iz.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jh.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jo.1.9 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jw.2.4 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.mu.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nc.2.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nl.2.3 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nt.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pg.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.po.2.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pt.2.10 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.px.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-120.dn.2.38 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.dj.1.3 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.bt.2.22 | $120$ | $5$ | $5$ | $16$ |