Invariants
| Level: | $120$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot3^{2}\cdot4^{2}\cdot6\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| 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: | 12J0 |
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
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(12)$ | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 120.96.1.lg.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qh.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qu.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.qz.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ra.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rc.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rd.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rg.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rh.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rj.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rm.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rn.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rq.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rr.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rs.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ru.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rv.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.rx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ry.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sa.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sb.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.se.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sf.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.si.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.sj.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.tb.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.te.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.tf.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ti.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.3.sa.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sd.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.se.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sh.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.sz.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.ta.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.td.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.te.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tg.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tj.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tk.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tn.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tp.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tq.3 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tt.4 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.tu.4 | $120$ | $2$ | $2$ | $3$ |
| 120.144.3.g.2 | $120$ | $3$ | $3$ | $3$ |
| 120.240.16.fn.2 | $120$ | $5$ | $5$ | $16$ |
| 120.288.15.ekn.4 | $120$ | $6$ | $6$ | $15$ |