Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&62\\108&55\end{bmatrix}$, $\begin{bmatrix}5&150\\110&19\end{bmatrix}$, $\begin{bmatrix}15&34\\64&53\end{bmatrix}$, $\begin{bmatrix}23&36\\140&83\end{bmatrix}$, $\begin{bmatrix}151&144\\98&43\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.24.0.w.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $3096576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-4.a.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 28.24.0-4.a.1.2 | $28$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.1-168.b.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.d.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.e.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.h.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.j.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.l.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.o.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.z.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bc.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bf.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bg.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bj.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bk.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bn.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bo.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.br.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.144.4-168.en.1.15 | $168$ | $3$ | $3$ | $4$ |
| 168.192.3-168.ez.1.22 | $168$ | $4$ | $4$ | $3$ |
| 168.384.11-168.cy.1.25 | $168$ | $8$ | $8$ | $11$ |