Invariants
| Level: | $156$ | $\SL_2$-level: | $156$ | Newform level: | $1$ | ||
| Index: | $336$ | $\PSL_2$-index: | $168$ | ||||
| Genus: | $11 = 1 + \frac{ 168 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6\cdot13\cdot26\cdot39\cdot78$ | Cusp orbits | $1^{8}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 11$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 11$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 78C11 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}100&119\\51&116\end{bmatrix}$, $\begin{bmatrix}101&0\\20&55\end{bmatrix}$, $\begin{bmatrix}118&93\\63&70\end{bmatrix}$, $\begin{bmatrix}125&58\\12&67\end{bmatrix}$, $\begin{bmatrix}133&122\\30&17\end{bmatrix}$, $\begin{bmatrix}135&92\\62&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 78.168.11.a.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $2$ |
| Cyclic 156-torsion field degree: | $96$ |
| Full 156-torsion field degree: | $359424$ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $14$ | $14$ | $0$ | $0$ |
| $X_0(13)$ | $13$ | $24$ | $12$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0-6.a.1.6 | $12$ | $14$ | $14$ | $0$ | $0$ |