Modular curve 168.192.5.kp.2

• Label: 168.192.5.kp.2
• Level: $168$
• Index: $192$
• Genus: $5$
• Cusps: $24$
• $\Q$-cusps: $0$

Invariants

Level:	$168$		$\SL_2$-level:	$12$		Newform level:	$1$
Index:	$192$		$\PSL_2$-index:	$192$
Genus:	$5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps:	$24$ (none of which are rational)		Cusp widths	$4^{12}\cdot12^{12}$		Cusp orbits	$2^{2}\cdot4^{5}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12E5

Level structure

$\GL_2(\Z/168\Z)$-generators:	$\begin{bmatrix}11&124\\114&61\end{bmatrix}$, $\begin{bmatrix}21&134\\50&9\end{bmatrix}$, $\begin{bmatrix}37&126\\48&97\end{bmatrix}$, $\begin{bmatrix}107&162\\146&55\end{bmatrix}$, $\begin{bmatrix}111&148\\122&55\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	168.384.5-168.kp.2.1, 168.384.5-168.kp.2.2, 168.384.5-168.kp.2.3, 168.384.5-168.kp.2.4, 168.384.5-168.kp.2.5, 168.384.5-168.kp.2.6, 168.384.5-168.kp.2.7, 168.384.5-168.kp.2.8, 168.384.5-168.kp.2.9, 168.384.5-168.kp.2.10, 168.384.5-168.kp.2.11, 168.384.5-168.kp.2.12, 168.384.5-168.kp.2.13, 168.384.5-168.kp.2.14, 168.384.5-168.kp.2.15, 168.384.5-168.kp.2.16
Cyclic 168-isogeny field degree:	$32$
Cyclic 168-torsion field degree:	$768$
Full 168-torsion field degree:	$774144$

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

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
24.96.1.cp.2	$24$	$2$	$2$	$1$	$1$
84.96.3.r.1	$84$	$2$	$2$	$3$	$?$
168.96.1.lm.2	$168$	$2$	$2$	$1$	$?$
168.96.1.lw.3	$168$	$2$	$2$	$1$	$?$
168.96.3.dm.1	$168$	$2$	$2$	$3$	$?$
168.96.3.el.2	$168$	$2$	$2$	$3$	$?$
168.96.3.eo.2	$168$	$2$	$2$	$3$	$?$