Invariants
| Level: | $156$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $4^{12}\cdot12^{12}$ | Cusp orbits | $2^{4}\cdot4^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E5 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}5&40\\30&151\end{bmatrix}$, $\begin{bmatrix}31&132\\142&47\end{bmatrix}$, $\begin{bmatrix}147&46\\34&81\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.192.5.o.3 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $28$ |
| Cyclic 156-torsion field degree: | $672$ |
| Full 156-torsion field degree: | $314496$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.192.1-12.d.1.8 | $12$ | $2$ | $2$ | $1$ | $0$ |
| 156.192.1-12.d.1.2 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 156.192.1-156.f.4.3 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 156.192.1-156.f.4.16 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 156.192.1-156.g.4.2 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 156.192.1-156.g.4.16 | $156$ | $2$ | $2$ | $1$ | $?$ |
| 156.192.3-156.j.1.2 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.j.1.11 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.n.1.4 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.n.1.5 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.p.2.4 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.p.2.11 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.q.2.6 | $156$ | $2$ | $2$ | $3$ | $?$ |
| 156.192.3-156.q.2.15 | $156$ | $2$ | $2$ | $3$ | $?$ |