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: | $\begin{bmatrix}51&40\\76&3\end{bmatrix}$, $\begin{bmatrix}111&118\\106&129\end{bmatrix}$, $\begin{bmatrix}125&28\\62&63\end{bmatrix}$, $\begin{bmatrix}149&85\\128&123\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 156.48.1.bq.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.k.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 |
| 78.48.0-78.b.1.3 | $78$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-78.b.1.11 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-156.p.1.11 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-156.p.1.12 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.1-12.l.1.1 | $156$ | $2$ | $2$ | $1$ | $?$ | 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.bt.1.7 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.bu.1.8 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.bv.1.7 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.192.3-156.bw.1.8 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.288.5-156.fk.1.1 | $156$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 312.192.3-312.vo.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vp.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vw.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.vx.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.we.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wf.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wg.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wh.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wq.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wr.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wu.1.26 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.wv.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.by.1.32 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.ca.1.31 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.ge.1.32 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.gg.1.31 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.pg.1.31 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.pi.1.32 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.qm.1.31 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.qo.1.32 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |