Invariants
Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (of which $4$ are rational) | Cusp widths | $8^{12}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot4$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $4$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8B3 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}71&80\\108&131\end{bmatrix}$, $\begin{bmatrix}103&160\\148&47\end{bmatrix}$, $\begin{bmatrix}113&44\\8&55\end{bmatrix}$, $\begin{bmatrix}127&92\\92&143\end{bmatrix}$, $\begin{bmatrix}145&80\\84&89\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 168.96.3.co.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $1536$ |
Full 168-torsion field degree: | $774144$ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.96.0-8.c.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
56.96.0-8.c.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ |
168.96.1-168.n.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.1-168.n.1.13 | $168$ | $2$ | $2$ | $1$ | $?$ |
168.96.2-168.a.1.7 | $168$ | $2$ | $2$ | $2$ | $?$ |
168.96.2-168.a.1.25 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
168.384.5-168.hp.1.1 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hp.1.15 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hp.2.6 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hp.2.15 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hq.1.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hq.1.16 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hq.2.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hq.2.13 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hu.1.1 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hu.1.15 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hu.2.6 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hu.2.15 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hv.3.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hv.3.16 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hv.4.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.hv.4.13 | $168$ | $2$ | $2$ | $5$ |