Invariants
| Level: | $120$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}7&96\\100&67\end{bmatrix}$, $\begin{bmatrix}37&8\\20&93\end{bmatrix}$, $\begin{bmatrix}61&58\\74&79\end{bmatrix}$, $\begin{bmatrix}83&78\\88&19\end{bmatrix}$, $\begin{bmatrix}115&106\\102&73\end{bmatrix}$, $\begin{bmatrix}119&4\\70&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 20.24.0.b.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| Full 120-torsion field degree: | $737280$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3^3}{5^2}\cdot\frac{(x-8y)^{24}(x^{4}-20x^{3}y+360x^{2}y^{2}-800xy^{3}+1600y^{4})^{3}(7x^{4}-20x^{3}y-120x^{2}y^{2}-800xy^{3}+11200y^{4})^{3}}{(x-8y)^{24}(x^{2}-10xy-20y^{2})^{4}(x^{2}-4xy+40y^{2})^{4}(x^{2}+20xy-80y^{2})^{4}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.240.8-20.d.1.2 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-20.d.1.13 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-20.d.1.1 | $120$ | $10$ | $10$ | $15$ |
| 120.96.0-40.g.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.g.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.g.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.g.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.h.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.h.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.h.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.h.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-40.j.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.j.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.k.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.k.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bp.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bp.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bq.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-40.bq.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-40.b.1.4 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.b.1.7 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.b.1.12 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.b.1.15 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.c.1.2 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.c.1.8 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.c.1.10 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-40.c.1.16 | $120$ | $2$ | $2$ | $2$ |
| 120.144.4-60.b.1.38 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-60.b.1.5 | $120$ | $4$ | $4$ | $3$ |
| 120.96.0-120.p.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.p.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.p.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.p.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.q.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.q.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.q.1.15 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.q.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.bk.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bk.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bm.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.bm.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.eu.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.eu.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ew.1.15 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.ew.1.16 | $120$ | $2$ | $2$ | $1$ |
| 120.96.2-120.c.1.25 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.c.1.26 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.c.1.31 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.c.1.32 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.d.1.25 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.d.1.26 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.d.1.31 | $120$ | $2$ | $2$ | $2$ |
| 120.96.2-120.d.1.32 | $120$ | $2$ | $2$ | $2$ |