Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{2}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8K1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}47&148\\104&47\end{bmatrix}$, $\begin{bmatrix}59&36\\0&125\end{bmatrix}$, $\begin{bmatrix}97&128\\100&153\end{bmatrix}$, $\begin{bmatrix}111&124\\28&69\end{bmatrix}$, $\begin{bmatrix}145&148\\144&107\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.96.1.a.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $774144$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + y^{2} + 2 z^{2} $ |
| $=$ | $3 x^{2} - 3 y^{2} - w^{2}$ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{(36z^{4}-36z^{3}w+18z^{2}w^{2}-6zw^{3}+w^{4})^{3}(36z^{4}+36z^{3}w+18z^{2}w^{2}+6zw^{3}+w^{4})^{3}}{w^{8}z^{8}(6z^{2}-w^{2})^{2}(6z^{2}+w^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.96.0-8.a.1.10 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.96.0-8.a.1.3 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-24.b.2.1 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-24.b.2.16 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-24.p.1.5 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-24.p.1.12 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-24.q.2.5 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-24.q.2.16 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.1-24.n.2.11 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.n.2.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bg.2.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bg.2.16 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bh.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.bh.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 168.384.5-24.i.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-24.k.2.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-24.l.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-24.n.4.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.p.2.13 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.q.2.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.u.2.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.w.2.9 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |