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: | $\begin{bmatrix}39&116\\116&3\end{bmatrix}$, $\begin{bmatrix}40&91\\119&64\end{bmatrix}$, $\begin{bmatrix}65&62\\74&109\end{bmatrix}$, $\begin{bmatrix}83&10\\80&69\end{bmatrix}$, $\begin{bmatrix}94&113\\105&46\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.12.0.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, including 1242 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^4}\cdot\frac{(3x+y)^{12}(9x^{4}+12x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{8}(3x+y)^{12}(12x^{2}+y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-4.c.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.12.0-4.c.1.6 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-24.m.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.o.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.v.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.w.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bl.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bm.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bo.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.br.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cn.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cp.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cr.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ct.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dl.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dn.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dp.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dr.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-24.cl.1.14 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-24.it.1.13 | $120$ | $4$ | $4$ | $1$ |
| 120.120.4-120.bx.1.11 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-120.byf.1.44 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-120.cv.1.52 | $120$ | $10$ | $10$ | $7$ |