Invariants
| Level: | $156$ | $\SL_2$-level: | $12$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}41&27\\29&28\end{bmatrix}$, $\begin{bmatrix}81&109\\29&58\end{bmatrix}$, $\begin{bmatrix}90&59\\127&143\end{bmatrix}$, $\begin{bmatrix}100&69\\21&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.8.0.a.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $84$ |
| Cyclic 156-torsion field degree: | $4032$ |
| Full 156-torsion field degree: | $7547904$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 224 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{8}(x^{2}+12y^{2})^{3}(x^{2}+108y^{2})}{y^{6}x^{10}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 156.4.0-2.a.1.1 | $156$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.32.0-12.a.1.3 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-12.a.1.4 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-12.a.2.1 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-12.a.2.2 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-156.a.1.1 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-156.a.1.5 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-156.a.2.3 | $156$ | $2$ | $2$ | $0$ |
| 156.32.0-156.a.2.7 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-6.a.1.10 | $156$ | $3$ | $3$ | $0$ |
| 156.48.0-78.a.1.2 | $156$ | $3$ | $3$ | $0$ |
| 156.48.0-78.a.1.6 | $156$ | $3$ | $3$ | $0$ |
| 156.48.1-6.a.1.3 | $156$ | $3$ | $3$ | $1$ |
| 156.64.1-12.a.1.1 | $156$ | $4$ | $4$ | $1$ |
| 156.224.7-78.a.1.3 | $156$ | $14$ | $14$ | $7$ |
| 312.32.0-24.a.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-24.a.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-24.a.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-24.a.2.7 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-312.a.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-312.a.1.13 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-312.a.2.7 | $312$ | $2$ | $2$ | $0$ |
| 312.32.0-312.a.2.15 | $312$ | $2$ | $2$ | $0$ |