Invariants
| Level: | $156$ | $\SL_2$-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 |
|---|---|---|---|---|
| 156.96.1.b.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.i.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.j.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.k.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.l.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.m.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.n.1 | $156$ | $2$ | $2$ | $1$ |
| 156.96.1.o.1 | $156$ | $2$ | $2$ | $1$ |
| 156.144.3.c.1 | $156$ | $3$ | $3$ | $3$ |
| 312.96.1.qh.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.qu.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.qy.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ra.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rb.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.re.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rf.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ri.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rk.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rl.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ro.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rp.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rt.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rw.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.rz.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sc.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sd.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sh.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.sk.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.tc.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.td.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.tg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.th.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.3.sb.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sc.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sf.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sg.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.sy.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tb.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tc.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tf.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.th.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.ti.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tl.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tm.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.to.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tr.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.ts.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.tv.1 | $312$ | $2$ | $2$ | $3$ |