Invariants
| Level: | $156$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}29&90\\44&73\end{bmatrix}$, $\begin{bmatrix}95&0\\44&139\end{bmatrix}$, $\begin{bmatrix}127&83\\78&5\end{bmatrix}$, $\begin{bmatrix}149&0\\146&127\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.48.1.br.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $28$ |
| Cyclic 156-torsion field degree: | $1344$ |
| Full 156-torsion field degree: | $1257984$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | not computed |
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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 52.12.0.l.1 | $52$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 156.48.0-156.o.1.9 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-156.o.1.14 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-156.q.1.11 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-156.q.1.14 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.1-12.l.1.5 | $156$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 156.192.3-156.bx.1.8 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.by.1.8 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.bz.1.6 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.ca.1.8 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.288.5-156.fl.1.1 | $156$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 312.192.3-312.vs.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vt.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wa.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wb.1.19 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wy.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wz.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.xa.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.xb.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.xk.1.19 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.xl.1.27 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.xo.1.18 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.xp.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.cc.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.ce.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.gi.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.gk.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.pk.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.pm.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.qq.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.qs.1.30 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |