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: | $\begin{bmatrix}19&64\\12&79\end{bmatrix}$, $\begin{bmatrix}43&96\\26&89\end{bmatrix}$, $\begin{bmatrix}121&16\\47&111\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.12.0.h.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $56$ |
| Cyclic 156-torsion field degree: | $2688$ |
| 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 771 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3}\cdot\frac{(3x-2y)^{12}(9x^{4}+42x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(3x-2y)^{12}(3x^{2}-y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 52.12.0-4.c.1.1 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 156.12.0-4.c.1.2 | $156$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 156.72.2-12.t.1.2 | $156$ | $3$ | $3$ | $2$ |
| 156.96.1-12.l.1.2 | $156$ | $4$ | $4$ | $1$ |
| 156.336.11-156.p.1.6 | $156$ | $14$ | $14$ | $11$ |
| 312.48.0-24.bk.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bk.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bl.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bl.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bs.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bs.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bt.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-24.bt.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cc.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cc.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cd.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cd.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cg.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.cg.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ch.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ch.1.12 | $312$ | $2$ | $2$ | $0$ |