Invariants
| Level: | $156$ | $\SL_2$-level: | $6$ | Newform level: | $36$ | ||
| 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 | $1^{2}\cdot2^{5}$ | ||
| 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: | $\begin{bmatrix}1&78\\84&61\end{bmatrix}$, $\begin{bmatrix}17&108\\66&143\end{bmatrix}$, $\begin{bmatrix}36&127\\127&42\end{bmatrix}$, $\begin{bmatrix}60&155\\47&30\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.72.1.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: | 36.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 27 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^3}\cdot\frac{(y-9z)^{3}(y+9z)^{3}(y^{3}-81y^{2}z+243yz^{2}-2187z^{3})^{3}(y^{3}+81y^{2}z+243yz^{2}+2187z^{3})^{3}}{z^{2}y^{6}(y^{2}+27z^{2})^{2}(y^{2}+243z^{2})^{6}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 156.48.0-6.b.1.1 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 156.48.0-6.b.1.2 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 156.48.0-6.c.1.1 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 156.48.0-6.c.1.2 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 156.48.1-6.b.1.1 | $156$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 156.72.0-6.a.1.1 | $156$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 156.72.0-6.a.1.4 | $156$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 156.288.5-12.r.1.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-12.v.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-12.z.1.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-12.bd.1.2 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.em.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.eq.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.ga.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 156.288.5-156.ge.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.gd.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.hf.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.ib.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-24.jd.1.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bjk.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bkm.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bua.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bvc.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |