Invariants
| Level: | $168$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}13&30\\79&53\end{bmatrix}$, $\begin{bmatrix}45&125\\34&29\end{bmatrix}$, $\begin{bmatrix}149&124\\107&129\end{bmatrix}$, $\begin{bmatrix}157&5\\7&96\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.8.0.b.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $96$ |
| Cyclic 168-torsion field degree: | $4608$ |
| Full 168-torsion field degree: | $9289728$ |
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 |
|---|---|---|---|---|---|
| 24.8.0-3.a.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 168.8.0-3.a.1.5 | $168$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.48.0-84.p.1.7 | $168$ | $3$ | $3$ | $0$ |
| 168.48.1-84.g.1.4 | $168$ | $3$ | $3$ | $1$ |
| 168.64.1-84.d.1.6 | $168$ | $4$ | $4$ | $1$ |
| 168.128.3-84.f.1.12 | $168$ | $8$ | $8$ | $3$ |
| 168.336.12-84.f.1.16 | $168$ | $21$ | $21$ | $12$ |
| 168.448.15-84.f.1.6 | $168$ | $28$ | $28$ | $15$ |