Invariants
| Level: | $168$ | $\SL_2$-level: | $56$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot7^{2}\cdot8\cdot14\cdot56$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 5$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 5$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56D5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}74&25\\89&150\end{bmatrix}$, $\begin{bmatrix}81&4\\34&163\end{bmatrix}$, $\begin{bmatrix}89&40\\100&85\end{bmatrix}$, $\begin{bmatrix}99&152\\68&43\end{bmatrix}$, $\begin{bmatrix}147&50\\8&77\end{bmatrix}$, $\begin{bmatrix}152&81\\81&40\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.5.gc.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $8$ |
| Cyclic 168-torsion field degree: | $192$ |
| Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ |
| 24.24.0-24.ba.1.12 | $24$ | $8$ | $8$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-24.ba.1.12 | $24$ | $8$ | $8$ | $0$ | $0$ |
| 56.96.2-28.c.1.25 | $56$ | $2$ | $2$ | $2$ | $0$ |
| 84.96.2-28.c.1.4 | $84$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.