Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot8$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 8C0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}14&41\\43&84\end{bmatrix}$, $\begin{bmatrix}58&71\\107&86\end{bmatrix}$, $\begin{bmatrix}64&113\\83&30\end{bmatrix}$, $\begin{bmatrix}75&44\\34&93\end{bmatrix}$, $\begin{bmatrix}79&96\\62&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.12.0.bb.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $48$ |
| Cyclic 120-torsion field degree: | $1536$ |
| Full 120-torsion field degree: | $1474560$ |
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 |
|---|---|---|---|---|---|
| 24.12.0-4.c.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 40.12.0-4.c.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-120.x.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ba.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.br.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bs.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bu.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bx.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ch.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ci.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cm.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cp.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cz.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.da.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dc.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.df.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ef.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eg.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-120.dh.1.18 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-120.zz.1.46 | $120$ | $4$ | $4$ | $1$ |
| 120.120.4-120.cb.1.25 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-120.byn.1.40 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-120.dh.1.34 | $120$ | $10$ | $10$ | $7$ |