Invariants
| Level: | $84$ | $\SL_2$-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 |
|---|---|---|---|---|
| 84.96.1.b.3 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.i.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.j.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.k.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.l.3 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.m.3 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.n.3 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1.o.2 | $84$ | $2$ | $2$ | $1$ |
| 84.144.3.c.1 | $84$ | $3$ | $3$ | $3$ |
| 84.384.23.j.1 | $84$ | $8$ | $8$ | $23$ |
| 168.96.1.qh.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.qu.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.qw.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.qy.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.qz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rc.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rd.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rg.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ri.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rj.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rm.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rn.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rr.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ru.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.rx.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sa.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sb.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.se.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.sf.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.si.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ta.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.tb.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.te.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.tf.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.3.pn.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.po.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.pr.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.ps.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qk.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qn.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qo.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qr.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qt.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qu.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qx.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.qy.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.ra.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.rd.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.re.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.rh.1 | $168$ | $2$ | $2$ | $3$ |
| 252.144.3.c.3 | $252$ | $3$ | $3$ | $3$ |
| 252.144.8.e.3 | $252$ | $3$ | $3$ | $8$ |
| 252.144.8.f.2 | $252$ | $3$ | $3$ | $8$ |