Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| 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 | $3^{8}\cdot12^{4}$ | 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: | 12S1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}35&114\\36&149\end{bmatrix}$, $\begin{bmatrix}73&78\\51&1\end{bmatrix}$, $\begin{bmatrix}131&78\\38&25\end{bmatrix}$, $\begin{bmatrix}149&162\\132&41\end{bmatrix}$, $\begin{bmatrix}157&156\\93&121\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.1.t.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $1032192$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 8 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{(y^{2}+24z^{2})^{3}(y^{6}+72y^{4}z^{2}+15552y^{2}z^{4}+124416z^{6})^{3}}{z^{4}y^{12}(y^{2}+8z^{2})(y^{2}+72z^{2})^{3}}$ |
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 |
|---|---|---|---|---|---|---|
| 8.6.0.d.1 | $8$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 21.24.0-3.a.1.1 | $21$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.72.0-6.a.1.1 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.48.0-24.bx.1.13 | $168$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 168.48.0-24.bx.1.14 | $168$ | $3$ | $3$ | $0$ | $?$ | full Jacobian |
| 168.72.0-6.a.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 168.288.5-24.d.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.bs.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.dn.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.ds.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.gt.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.gu.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.hd.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-24.hf.1.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.od.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.oe.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ok.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ol.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.rn.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ro.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.ru.1.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.rv.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |