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: | $\begin{bmatrix}7&12\\32&83\end{bmatrix}$, $\begin{bmatrix}15&92\\28&117\end{bmatrix}$, $\begin{bmatrix}49&4\\102&55\end{bmatrix}$, $\begin{bmatrix}79&76\\16&23\end{bmatrix}$, $\begin{bmatrix}81&20\\8&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.48.0.bg.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.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.e.1.5 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.48.0-60.c.1.10 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-60.c.1.21 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.u.1.23 | $120$ | $2$ | $2$ | $0$ | $?$ |
| 120.48.0-120.u.1.54 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.cd.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.da.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ea.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.eg.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jd.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.jf.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.js.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.ju.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.my.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.na.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.np.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.nr.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pi.1.10 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.po.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.pu.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.192.1-120.px.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.288.8-120.dr.2.60 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.dl.1.9 | $120$ | $4$ | $4$ | $7$ |
| 120.480.16-120.bv.2.15 | $120$ | $5$ | $5$ | $16$ |