Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 96$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24J1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}16&51\\95&128\end{bmatrix}$, $\begin{bmatrix}48&149\\133&64\end{bmatrix}$, $\begin{bmatrix}106&121\\129&14\end{bmatrix}$, $\begin{bmatrix}125&132\\78&107\end{bmatrix}$, $\begin{bmatrix}167&50\\72&121\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 168.96.1.tg.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $384$ |
| Full 168-torsion field degree: | $774144$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | not computed |
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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.p.1.7 | $8$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 21.8.0-3.a.1.2 | $21$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.ix.1.22 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 168.96.0-168.dp.1.50 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-168.dp.1.52 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-168.dq.1.13 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.0-168.dq.1.36 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
| 168.96.1-24.ix.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 168.384.5-168.qv.4.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.rs.3.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vf.2.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.vi.2.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.xh.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.xm.1.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.yv.4.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.za.2.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhk.4.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bhq.4.11 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bia.3.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.big.2.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bjw.2.3 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkc.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bkm.4.4 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.bks.3.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |