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: | 24B5 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}4&149\\119&144\end{bmatrix}$, $\begin{bmatrix}30&139\\71&162\end{bmatrix}$, $\begin{bmatrix}58&59\\29&38\end{bmatrix}$, $\begin{bmatrix}68&3\\165&104\end{bmatrix}$, $\begin{bmatrix}106&165\\69&118\end{bmatrix}$, $\begin{bmatrix}107&54\\74&121\end{bmatrix}$ |
| 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.cx.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.36.1.gl.1 | $24$ | $2$ | $2$ | $1$ | $1$ |
| 56.24.1.cx.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
| 84.36.1.fh.1 | $84$ | $2$ | $2$ | $1$ | $?$ |
| 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.baz.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.bql.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.eai.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ear.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.jzv.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.kad.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ked.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.kel.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qgz.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qhd.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qjl.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qjp.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qqv.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qqz.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qth.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.qtl.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.sph.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.spl.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.srt.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.srx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.szd.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.szh.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.tbp.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.tbt.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.vof.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.voj.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.vqr.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.vqv.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.vyb.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.vyf.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.wan.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.war.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ybl.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ybp.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ydx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.yeb.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ylt.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.ylx.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.yoj.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.yon.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.baph.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.bapp.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.basy.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.batg.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.bban.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.bbar.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.bbee.1 | $168$ | $2$ | $2$ | $9$ |
| 168.144.9.bbei.1 | $168$ | $2$ | $2$ | $9$ |