Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24V3 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}89&142\\96&67\end{bmatrix}$, $\begin{bmatrix}93&26\\104&93\end{bmatrix}$, $\begin{bmatrix}101&18\\96&137\end{bmatrix}$, $\begin{bmatrix}109&161\\132&41\end{bmatrix}$, $\begin{bmatrix}137&150\\48&65\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.3.ms.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $384$ |
| Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has no real points, and therefore no 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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ |
| 56.48.0-56.bs.1.6 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.96.1-24.iw.1.31 | $24$ | $2$ | $2$ | $1$ | $0$ |
| 56.48.0-56.bs.1.6 | $56$ | $4$ | $4$ | $0$ | $0$ |
| 168.96.1-24.iw.1.28 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zd.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zd.1.30 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zz.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ |
| 168.96.1-168.zz.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.384.5-168.bbh.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbh.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbh.3.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbh.4.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbl.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbl.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbl.3.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bbl.4.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bib.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bib.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bib.3.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bib.4.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bif.1.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bif.2.10 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bif.3.4 | $168$ | $2$ | $2$ | $5$ |
| 168.384.5-168.bif.4.4 | $168$ | $2$ | $2$ | $5$ |