Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 12E0 |
Level structure
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ |
| 40.6.0.d.1 | $40$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
| 40.6.0.d.1 | $40$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.48.1.dh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.gl.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.kf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.kh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.baf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bah.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bal.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.ban.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bza.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzi.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzk.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzr.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bzt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.72.1.cl.1 | $120$ | $3$ | $3$ | $1$ |
| 120.120.8.jb.1 | $120$ | $5$ | $5$ | $8$ |
| 120.144.7.hmo.1 | $120$ | $6$ | $6$ | $7$ |
| 120.240.15.bix.1 | $120$ | $10$ | $10$ | $15$ |