Invariants
| Level: | $156$ | $\SL_2$-level: | $4$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| 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: | 4E0 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}11&148\\126&119\end{bmatrix}$, $\begin{bmatrix}31&104\\26&33\end{bmatrix}$, $\begin{bmatrix}49&126\\124&101\end{bmatrix}$, $\begin{bmatrix}121&10\\114&85\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 52.12.0.b.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $112$ |
| Cyclic 156-torsion field degree: | $5376$ |
| Full 156-torsion field degree: | $5031936$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 358 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^4\cdot13^2}\cdot\frac{(13x+12y)^{12}(169x^{4}-117x^{2}y^{2}+81y^{4})^{3}}{y^{4}x^{4}(13x+12y)^{12}(13x^{2}-9y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.12.0-2.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 156.12.0-2.a.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 |
|---|---|---|---|---|
| 156.48.0-52.b.1.2 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-52.c.1.3 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-52.c.1.4 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.e.1.1 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.e.1.4 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.g.1.2 | $156$ | $2$ | $2$ | $0$ |
| 156.48.0-156.g.1.3 | $156$ | $2$ | $2$ | $0$ |
| 156.72.2-156.b.1.4 | $156$ | $3$ | $3$ | $2$ |
| 156.96.1-156.b.1.7 | $156$ | $4$ | $4$ | $1$ |
| 156.336.11-52.d.1.5 | $156$ | $14$ | $14$ | $11$ |
| 312.48.0-104.d.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.d.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.g.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.g.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.l.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.l.1.10 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.r.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.r.1.10 | $312$ | $2$ | $2$ | $0$ |