Invariants
| Level: | $120$ | $\SL_2$-level: | $4$ | ||||
| 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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
Level structure
| $\GL_2(\Z/120\Z)$-generators: | $\begin{bmatrix}13&100\\80&91\end{bmatrix}$, $\begin{bmatrix}31&102\\50&97\end{bmatrix}$, $\begin{bmatrix}43&84\\108&43\end{bmatrix}$, $\begin{bmatrix}59&18\\82&7\end{bmatrix}$, $\begin{bmatrix}77&36\\22&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.12.0.a.1 for the level structure with $-I$) |
| Cyclic 120-isogeny field degree: | $96$ |
| Cyclic 120-torsion field degree: | $3072$ |
| 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 771 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^2}\cdot\frac{(3x+y)^{12}(12x^{2}-6xy+y^{2})^{3}(12x^{2}+6xy+y^{2})^{3}}{y^{4}x^{4}(3x+y)^{12}(12x^{2}-y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-2.a.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 120.12.0-2.a.1.1 | $120$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-12.a.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.a.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.c.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-12.c.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-12.c.1.8 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-12.c.1.8 | $120$ | $4$ | $4$ | $1$ |
| 120.48.0-24.b.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.b.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.f.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.f.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.d.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.d.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.e.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.e.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.120.4-60.a.1.5 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-60.a.1.1 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-60.a.1.7 | $120$ | $10$ | $10$ | $7$ |
| 120.48.0-120.h.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.h.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.k.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.k.1.12 | $120$ | $2$ | $2$ | $0$ |