Invariants
| Level: | $156$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $3 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12D3 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}41&36\\150&149\end{bmatrix}$, $\begin{bmatrix}47&44\\144&145\end{bmatrix}$, $\begin{bmatrix}49&108\\24&79\end{bmatrix}$, $\begin{bmatrix}77&113\\150&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.72.3.mb.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $28$ |
| Cyclic 156-torsion field degree: | $1344$ |
| Full 156-torsion field degree: | $838656$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.72.0-6.a.1.6 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 156.48.1-156.p.1.4 | $156$ | $3$ | $3$ | $1$ | $?$ |
| 156.48.1-156.p.1.8 | $156$ | $3$ | $3$ | $1$ | $?$ |
| 156.72.0-6.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 |
|---|---|---|---|---|
| 156.288.5-156.a.1.6 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.bg.1.3 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.bw.1.2 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.cb.1.6 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.fp.1.4 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.fs.1.2 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.fy.1.3 | $156$ | $2$ | $2$ | $5$ |
| 156.288.5-156.ge.1.4 | $156$ | $2$ | $2$ | $5$ |
| 312.288.5-312.eb.1.8 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.iu.1.2 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.nc.1.6 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.ol.1.6 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.bqu.1.4 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.brp.1.8 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.btf.1.3 | $312$ | $2$ | $2$ | $5$ |
| 312.288.5-312.buy.1.6 | $312$ | $2$ | $2$ | $5$ |