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: | $10$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 14$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 8$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 24B8 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}7&78\\80&119\end{bmatrix}$, $\begin{bmatrix}33&50\\148&165\end{bmatrix}$, $\begin{bmatrix}51&154\\92&69\end{bmatrix}$, $\begin{bmatrix}127&110\\80&143\end{bmatrix}$, $\begin{bmatrix}139&120\\36&115\end{bmatrix}$, $\begin{bmatrix}141&110\\124&93\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.144.8.nx.2 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $516096$ |
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 |
---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ |
56.96.0-56.v.1.12 | $56$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.144.4-24.z.2.47 | $24$ | $2$ | $2$ | $4$ | $0$ |
56.96.0-56.v.1.12 | $56$ | $3$ | $3$ | $0$ | $0$ |
168.144.4-24.z.2.32 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.bp.1.28 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.bp.1.128 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.et.1.36 | $168$ | $2$ | $2$ | $4$ | $?$ |
168.144.4-168.et.1.49 | $168$ | $2$ | $2$ | $4$ | $?$ |