Invariants
| Level: | $156$ | $\SL_2$-level: | $4$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}53&22\\111&125\end{bmatrix}$, $\begin{bmatrix}71&76\\6&77\end{bmatrix}$, $\begin{bmatrix}99&130\\76&153\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 156-isogeny field degree: | $112$ |
| Cyclic 156-torsion field degree: | $5376$ |
| Full 156-torsion field degree: | $10063872$ |
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 |
|---|---|---|---|---|---|
| 12.6.0.f.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 52.6.0.a.1 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 156.6.0.b.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 |
|---|---|---|---|---|
| 156.36.2.fb.1 | $156$ | $3$ | $3$ | $2$ |
| 156.48.1.bh.1 | $156$ | $4$ | $4$ | $1$ |
| 156.168.11.dj.1 | $156$ | $14$ | $14$ | $11$ |