Invariants
| Level: | $156$ | $\SL_2$-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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ |
| 52.6.0.c.1 | $52$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
| 52.6.0.c.1 | $52$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.48.1.b.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.h.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.r.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.t.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.bl.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.bn.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.bo.1 | $156$ | $2$ | $2$ | $1$ |
| 156.48.1.br.1 | $156$ | $2$ | $2$ | $1$ |
| 156.72.1.n.1 | $156$ | $3$ | $3$ | $1$ |
| 156.336.23.dd.1 | $156$ | $14$ | $14$ | $23$ |
| 312.48.1.gg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.kg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bag.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bam.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.byz.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bzf.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bzj.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bzs.1 | $312$ | $2$ | $2$ | $1$ |