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 $6$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{6}\cdot2^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $6$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 6F1 |
Level structure
$\GL_2(\Z/156\Z)$-generators: | $\begin{bmatrix}19&48\\60&131\end{bmatrix}$, $\begin{bmatrix}43&126\\84&47\end{bmatrix}$, $\begin{bmatrix}53&48\\114&137\end{bmatrix}$, $\begin{bmatrix}65&18\\102&19\end{bmatrix}$, $\begin{bmatrix}83&132\\12&73\end{bmatrix}$, $\begin{bmatrix}133&6\\12&101\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 6.72.1.a.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} + 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 $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{(y^{2}+3z^{2})^{3}(y^{6}+225y^{4}z^{2}-405y^{2}z^{4}+243z^{6})^{3}}{z^{2}y^{6}(y-3z)^{6}(y-z)^{2}(y+z)^{2}(y+3z)^{6}}$ |
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.a.1.1 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
156.48.0-6.a.1.5 | $156$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
156.48.1-6.a.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 |
---|---|---|---|---|---|---|
156.288.3-12.a.1.3 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
156.288.3-12.a.1.9 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
156.288.3-156.a.1.5 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
156.288.3-156.a.1.9 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
156.288.5-12.a.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-156.a.1.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-12.b.1.8 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-156.b.1.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-12.e.1.4 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-156.e.1.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-12.f.1.3 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.5-156.f.1.6 | $156$ | $2$ | $2$ | $5$ | $?$ | not computed |
156.288.7-12.o.1.6 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-12.o.1.7 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-156.o.1.9 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
156.288.7-156.o.1.13 | $156$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.3-24.a.1.4 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.288.3-24.a.1.17 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.288.3-312.a.1.11 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.288.3-312.a.1.17 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.288.5-24.a.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-312.a.1.9 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-24.d.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-312.d.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-24.m.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-312.m.1.4 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-24.p.1.6 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.5-312.p.1.10 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
312.288.7-312.co.1.17 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-312.co.1.26 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-24.cw.1.10 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |
312.288.7-24.cw.1.13 | $312$ | $2$ | $2$ | $7$ | $?$ | not computed |