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: | $\begin{bmatrix}1&112\\41&39\end{bmatrix}$, $\begin{bmatrix}43&8\\92&23\end{bmatrix}$, $\begin{bmatrix}53&108\\102&35\end{bmatrix}$, $\begin{bmatrix}101&72\\90&97\end{bmatrix}$, $\begin{bmatrix}119&88\\52&105\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.12.0.v.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 544 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^3}{5}\cdot\frac{(x-y)^{12}(25x^{4}+140x^{2}y^{2}+4y^{4})^{3}}{y^{2}x^{2}(x-y)^{12}(5x^{2}-2y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.12.0-4.c.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 60.12.0-4.c.1.1 | $60$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.0-40.bo.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bo.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bp.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bp.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.by.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.by.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bz.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bz.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cs.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.cs.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ct.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ct.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.da.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.da.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.db.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.db.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.72.2-120.cf.1.16 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-120.zf.1.9 | $120$ | $4$ | $4$ | $1$ |
| 120.120.4-40.bh.1.3 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-40.bt.1.3 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-40.cf.1.4 | $120$ | $10$ | $10$ | $7$ |