Invariants
| Level: | $156$ | $\SL_2$-level: | $4$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 4G0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}7&62\\0&83\end{bmatrix}$, $\begin{bmatrix}73&114\\22&5\end{bmatrix}$, $\begin{bmatrix}137&10\\4&99\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.24.0.g.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $112$ |
| Cyclic 156-torsion field degree: | $5376$ |
| Full 156-torsion field degree: | $2515968$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-12.b.1.4 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 52.24.0-52.b.1.4 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 156.24.0-156.a.1.3 | $156$ | $2$ | $2$ | $0$ | $?$ |
| 156.24.0-156.a.1.7 | $156$ | $2$ | $2$ | $0$ | $?$ |
| 156.24.0-12.b.1.2 | $156$ | $2$ | $2$ | $0$ | $?$ |
| 156.24.0-52.b.1.1 | $156$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.144.4-156.j.1.10 | $156$ | $3$ | $3$ | $4$ |
| 156.192.3-156.j.1.15 | $156$ | $4$ | $4$ | $3$ |