Modular curve 168.48.0.bh.2

• Label: 168.48.0.bh.2
• Level: $168$
• Index: $48$
• Genus: $0$
• Cusps: $10$
• $\Q$-cusps: $0$

Invariants

Level:	$168$		$\SL_2$-level:	$8$
Index:	$48$		$\PSL_2$-index:	$48$
Genus:	$0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$
Cusps:	$10$ (none of which are rational)		Cusp widths	$4^{8}\cdot8^{2}$		Cusp orbits	$2^{3}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
$\Q$-gonality:	$1 \le \gamma \le 2$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

8N0

Level structure

$\GL_2(\Z/168\Z)$-generators:	$\begin{bmatrix}7&20\\88&63\end{bmatrix}$, $\begin{bmatrix}23&48\\116&139\end{bmatrix}$, $\begin{bmatrix}31&86\\8&141\end{bmatrix}$, $\begin{bmatrix}59&32\\56&97\end{bmatrix}$, $\begin{bmatrix}77&68\\132&31\end{bmatrix}$, $\begin{bmatrix}121&86\\76&43\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	168.96.0-168.bh.2.1, 168.96.0-168.bh.2.2, 168.96.0-168.bh.2.3, 168.96.0-168.bh.2.4, 168.96.0-168.bh.2.5, 168.96.0-168.bh.2.6, 168.96.0-168.bh.2.7, 168.96.0-168.bh.2.8, 168.96.0-168.bh.2.9, 168.96.0-168.bh.2.10, 168.96.0-168.bh.2.11, 168.96.0-168.bh.2.12, 168.96.0-168.bh.2.13, 168.96.0-168.bh.2.14, 168.96.0-168.bh.2.15, 168.96.0-168.bh.2.16, 168.96.0-168.bh.2.17, 168.96.0-168.bh.2.18, 168.96.0-168.bh.2.19, 168.96.0-168.bh.2.20, 168.96.0-168.bh.2.21, 168.96.0-168.bh.2.22, 168.96.0-168.bh.2.23, 168.96.0-168.bh.2.24, 168.96.0-168.bh.2.25, 168.96.0-168.bh.2.26, 168.96.0-168.bh.2.27, 168.96.0-168.bh.2.28, 168.96.0-168.bh.2.29, 168.96.0-168.bh.2.30, 168.96.0-168.bh.2.31, 168.96.0-168.bh.2.32
Cyclic 168-isogeny field degree:	$64$
Cyclic 168-torsion field degree:	$3072$
Full 168-torsion field degree:	$3096576$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

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
8.24.0.d.1	$8$	$2$	$2$	$0$	$0$
168.24.0.e.1	$168$	$2$	$2$	$0$	$?$
168.24.0.u.1	$168$	$2$	$2$	$0$	$?$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus
168.96.1.d.1	$168$	$2$	$2$	$1$
168.96.1.ce.2	$168$	$2$	$2$	$1$
168.96.1.ei.2	$168$	$2$	$2$	$1$
168.96.1.eq.1	$168$	$2$	$2$	$1$
168.96.1.id.2	$168$	$2$	$2$	$1$
168.96.1.il.1	$168$	$2$	$2$	$1$
168.96.1.jy.1	$168$	$2$	$2$	$1$
168.96.1.kg.2	$168$	$2$	$2$	$1$
168.96.1.lz.1	$168$	$2$	$2$	$1$
168.96.1.mh.2	$168$	$2$	$2$	$1$
168.96.1.nu.2	$168$	$2$	$2$	$1$
168.96.1.oc.1	$168$	$2$	$2$	$1$
168.96.1.pb.2	$168$	$2$	$2$	$1$
168.96.1.pj.1	$168$	$2$	$2$	$1$
168.96.1.py.1	$168$	$2$	$2$	$1$
168.96.1.qc.2	$168$	$2$	$2$	$1$
168.144.8.do.2	$168$	$3$	$3$	$8$
168.192.7.di.2	$168$	$4$	$4$	$7$
168.384.23.cw.2	$168$	$8$	$8$	$23$