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: | $\begin{bmatrix}28&43\\81&74\end{bmatrix}$, $\begin{bmatrix}45&56\\22&35\end{bmatrix}$, $\begin{bmatrix}57&80\\70&27\end{bmatrix}$, $\begin{bmatrix}68&17\\33&44\end{bmatrix}$, $\begin{bmatrix}73&74\\4&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 120.12.0.z.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.1.6 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 120.12.0-4.c.1.2 | $120$ | $2$ | $2$ | $0$ | $?$ |
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.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ba.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bi.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bj.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.by.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cb.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cd.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ce.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cr.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cs.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cu.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cx.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dh.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.di.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dw.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dz.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-120.cx.1.62 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-120.zx.1.64 | $120$ | $4$ | $4$ | $1$ |
| 120.120.4-120.bz.1.31 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-120.byh.1.64 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-120.cx.1.64 | $120$ | $10$ | $10$ | $7$ |