Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | 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 | $4^{2}\cdot8^{2}$ | 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: | 8B1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}11&116\\160&123\end{bmatrix}$, $\begin{bmatrix}59&10\\144&145\end{bmatrix}$, $\begin{bmatrix}67&30\\28&29\end{bmatrix}$, $\begin{bmatrix}77&142\\116&107\end{bmatrix}$, $\begin{bmatrix}123&164\\116&1\end{bmatrix}$, $\begin{bmatrix}149&36\\140&85\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.d.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $3096576$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 36x $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{3888x^{2}y^{4}z^{2}+36xy^{6}z+5038848xy^{2}z^{5}+y^{8}+2176782336z^{8}}{z^{2}y^{4}x^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.24.0-4.b.1.11 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.24.0-4.b.1.10 | $168$ | $2$ | $2$ | $0$ | $?$ | 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.n.2.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bb.1.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bg.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bg.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bh.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bh.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bi.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bi.2.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bj.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bj.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bs.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bv.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bw.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.by.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ce.1.27 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ce.2.9 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cf.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cf.2.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cg.1.17 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cg.2.17 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ch.1.22 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ch.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.dc.1.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.de.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.144.5-24.h.1.29 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.192.5-24.h.1.8 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 168.384.13-168.d.1.41 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |