Invariants
Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $1$ | ||
Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 8$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 24B8 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}17&68\\80&21\end{bmatrix}$, $\begin{bmatrix}27&86\\100&69\end{bmatrix}$, $\begin{bmatrix}69&110\\104&133\end{bmatrix}$, $\begin{bmatrix}79&124\\100&137\end{bmatrix}$, $\begin{bmatrix}103&8\\164&41\end{bmatrix}$, $\begin{bmatrix}137&6\\160&73\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.144.8.ph.2 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $32$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $516096$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal 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 |
---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
56.96.0-56.ba.1.1 | $56$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ |
56.96.0-56.ba.1.1 | $56$ | $3$ | $3$ | $0$ | $0$ |
168.144.4-168.bi.2.10 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.bi.2.34 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.bp.1.38 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.bp.1.97 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-24.ch.1.31 | $168$ | $2$ | $2$ | $4$ | $?$ |