Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $5 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 5$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24A5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}19&146\\50&161\end{bmatrix}$, $\begin{bmatrix}25&5\\158&83\end{bmatrix}$, $\begin{bmatrix}83&13\\132&133\end{bmatrix}$, $\begin{bmatrix}87&166\\110&141\end{bmatrix}$, $\begin{bmatrix}111&85\\28&137\end{bmatrix}$, $\begin{bmatrix}127&80\\42&41\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $2064384$ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $24$ | $24$ | $0$ | $0$ |
| 56.24.1.o.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.36.2.bv.1 | $24$ | $2$ | $2$ | $2$ | $1$ |
| 56.24.1.o.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
| 168.36.0.ga.1 | $168$ | $2$ | $2$ | $0$ | $?$ |
| 168.36.3.c.1 | $168$ | $2$ | $2$ | $3$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.144.9.cim.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cin.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.crw.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.crx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ctw.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ctx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cwi.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cwj.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cxs.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cxu.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cyq.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.cyr.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.czq.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.czt.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.daq.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dar.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dca.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dcd.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ddw.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ddx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dft.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dfu.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dho.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dhp.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dka.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dkb.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dlx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dly.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dns.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dnt.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dpo.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dpr.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dqu.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dqv.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.drs.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dru.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dsq.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dsr.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dto.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dtq.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dvg.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dvh.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dxs.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dxt.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dzk.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.dzl.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.eaq.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ear.1 | $168$ | $2$ | $2$ | $9$ |