Invariants
| Level: | $156$ | $\SL_2$-level: | $6$ | Newform level: | $144$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F1 |
Level structure
| $\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}18&53\\29&48\end{bmatrix}$, $\begin{bmatrix}84&49\\109&96\end{bmatrix}$, $\begin{bmatrix}121&60\\102&73\end{bmatrix}$, $\begin{bmatrix}147&70\\142&63\end{bmatrix}$, $\begin{bmatrix}155&114\\144&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.72.1.b.1 for the level structure with $-I$) |
| Cyclic 156-isogeny field degree: | $28$ |
| Cyclic 156-torsion field degree: | $1344$ |
| Full 156-torsion field degree: | $838656$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 144.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 1 $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(y^{2}-3z^{2})^{3}(y^{6}-225y^{4}z^{2}-405y^{2}z^{4}-243z^{6})^{3}}{z^{2}y^{6}(y^{2}+z^{2})^{2}(y^{2}+9z^{2})^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 78.72.0-6.a.1.2 | $78$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.48.0-12.d.1.3 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 156.48.0-12.d.1.4 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 156.48.1-12.b.1.2 | $156$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 156.72.0-6.a.1.1 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 156.288.3-12.b.1.2 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.288.3-12.b.1.4 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.288.3-156.b.1.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.288.3-156.b.1.15 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.288.5-12.i.1.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-12.k.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-12.l.1.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-12.n.1.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.ba.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.bc.1.3 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.be.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.bg.1.3 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.7-12.bb.1.4 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-12.bb.1.6 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.el.1.5 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 156.288.7-156.el.1.11 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.3-24.b.1.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-24.b.1.2 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.b.1.10 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.3-312.b.1.22 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.288.5-24.cf.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.cs.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.cz.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.dn.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.ha.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.ho.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.ic.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.iq.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.7-24.or.1.9 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-24.or.1.10 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.bxh.1.12 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 312.288.7-312.bxh.1.24 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |