Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 9$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24C9 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}47&129\\4&37\end{bmatrix}$, $\begin{bmatrix}51&19\\140&45\end{bmatrix}$, $\begin{bmatrix}81&5\\164&7\end{bmatrix}$, $\begin{bmatrix}85&80\\72&29\end{bmatrix}$, $\begin{bmatrix}121&55\\52&155\end{bmatrix}$, $\begin{bmatrix}125&87\\0&59\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 168.288.9-168.ctf.1.1, 168.288.9-168.ctf.1.2, 168.288.9-168.ctf.1.3, 168.288.9-168.ctf.1.4, 168.288.9-168.ctf.1.5, 168.288.9-168.ctf.1.6, 168.288.9-168.ctf.1.7, 168.288.9-168.ctf.1.8, 168.288.9-168.ctf.1.9, 168.288.9-168.ctf.1.10, 168.288.9-168.ctf.1.11, 168.288.9-168.ctf.1.12, 168.288.9-168.ctf.1.13, 168.288.9-168.ctf.1.14, 168.288.9-168.ctf.1.15, 168.288.9-168.ctf.1.16 |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $1032192$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $48$ | $48$ | $0$ | $0$ |
| 56.48.1.dn.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.72.4.dw.1 | $24$ | $2$ | $2$ | $4$ | $1$ |
| 56.48.1.dn.1 | $56$ | $3$ | $3$ | $1$ | $1$ |
| 168.72.2.sy.1 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2.sz.1 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.4.ft.1 | $168$ | $2$ | $2$ | $4$ | $?$ |
| 168.72.5.bx.1 | $168$ | $2$ | $2$ | $5$ | $?$ |
| 168.72.5.zq.1 | $168$ | $2$ | $2$ | $5$ | $?$ |
| 168.72.5.zr.1 | $168$ | $2$ | $2$ | $5$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.288.17.qdd.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qdl.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qfp.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qfx.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qnd.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qnp.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qqb.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.qqj.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.rbt.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.rcb.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.ref.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.ren.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.rlp.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.rlx.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.rob.1 | $168$ | $2$ | $2$ | $17$ |
| 168.288.17.roj.1 | $168$ | $2$ | $2$ | $17$ |