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: | $\begin{bmatrix}6&77\\7&16\end{bmatrix}$, $\begin{bmatrix}18&109\\113&66\end{bmatrix}$, $\begin{bmatrix}85&24\\8&5\end{bmatrix}$, $\begin{bmatrix}103&82\\88&21\end{bmatrix}$, $\begin{bmatrix}119&60\\116&19\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.12.0.ba.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 |
|---|---|---|---|---|---|
| 8.12.0-4.c.1.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 120.12.0-4.c.1.4 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-120.y.1.22 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.z.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bq.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bs.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bv.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bw.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cg.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cj.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cn.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.co.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cy.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.db.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dd.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.de.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ee.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.eh.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-120.dg.1.47 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-120.zy.1.7 | $120$ | $4$ | $4$ | $1$ |
| 120.120.4-120.ca.1.30 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-120.bym.1.19 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-120.dg.1.9 | $120$ | $10$ | $10$ | $7$ |