Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}0&137\\97&164\end{bmatrix}$, $\begin{bmatrix}40&57\\43&116\end{bmatrix}$, $\begin{bmatrix}50&3\\135&128\end{bmatrix}$, $\begin{bmatrix}59&52\\60&127\end{bmatrix}$, $\begin{bmatrix}76&53\\133&144\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.eq.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $3096576$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 876x - 9520 $ |
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 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot3^6}\cdot\frac{24x^{2}y^{6}+392688x^{2}y^{4}z^{2}+1715447808x^{2}y^{2}z^{4}+2481592329216x^{2}z^{6}+1176xy^{6}z+13374720xy^{4}z^{3}+58582206720xy^{2}z^{5}+85595676880896xz^{7}+y^{8}+19968y^{6}z^{2}+146437632y^{4}z^{4}+524385073152y^{2}z^{6}+720863480254464z^{8}}{z^{4}y^{2}(48x^{2}+1632xz+y^{2}+13440z^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.24.0-6.a.1.15 | $168$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 168.96.1-24.bz.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.ci.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.dv.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.dw.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.ih.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.ii.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.ik.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.il.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byi.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byj.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byl.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bym.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byu.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byv.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byx.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.byy.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.144.3-24.qp.1.1 | $168$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 168.384.13-168.pc.1.48 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |