Modular curve 168.48.0.bp.1

• Label: 168.48.0.bp.1
• 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	$2^{4}\cdot4^{2}\cdot8^{4}$		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:

8O0

Level structure

$\GL_2(\Z/168\Z)$-generators:	$\begin{bmatrix}61&160\\134&33\end{bmatrix}$, $\begin{bmatrix}85&76\\52&131\end{bmatrix}$, $\begin{bmatrix}87&28\\148&55\end{bmatrix}$, $\begin{bmatrix}113&48\\162&121\end{bmatrix}$, $\begin{bmatrix}131&80\\104&13\end{bmatrix}$, $\begin{bmatrix}145&136\\132&35\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	168.96.0-168.bp.1.1, 168.96.0-168.bp.1.2, 168.96.0-168.bp.1.3, 168.96.0-168.bp.1.4, 168.96.0-168.bp.1.5, 168.96.0-168.bp.1.6, 168.96.0-168.bp.1.7, 168.96.0-168.bp.1.8, 168.96.0-168.bp.1.9, 168.96.0-168.bp.1.10, 168.96.0-168.bp.1.11, 168.96.0-168.bp.1.12, 168.96.0-168.bp.1.13, 168.96.0-168.bp.1.14, 168.96.0-168.bp.1.15, 168.96.0-168.bp.1.16, 168.96.0-168.bp.1.17, 168.96.0-168.bp.1.18, 168.96.0-168.bp.1.19, 168.96.0-168.bp.1.20, 168.96.0-168.bp.1.21, 168.96.0-168.bp.1.22, 168.96.0-168.bp.1.23, 168.96.0-168.bp.1.24, 168.96.0-168.bp.1.25, 168.96.0-168.bp.1.26, 168.96.0-168.bp.1.27, 168.96.0-168.bp.1.28, 168.96.0-168.bp.1.29, 168.96.0-168.bp.1.30, 168.96.0-168.bp.1.31, 168.96.0-168.bp.1.32
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.u.1	$168$	$2$	$2$	$0$	$?$
168.24.0.x.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.a.1	$168$	$2$	$2$	$1$
168.96.1.d.1	$168$	$2$	$2$	$1$
168.96.1.bz.2	$168$	$2$	$2$	$1$
168.96.1.ce.2	$168$	$2$	$2$	$1$
168.96.1.ds.2	$168$	$2$	$2$	$1$
168.96.1.dt.2	$168$	$2$	$2$	$1$
168.96.1.ea.1	$168$	$2$	$2$	$1$
168.96.1.eb.1	$168$	$2$	$2$	$1$
168.96.1.gg.2	$168$	$2$	$2$	$1$
168.96.1.gh.2	$168$	$2$	$2$	$1$
168.96.1.go.1	$168$	$2$	$2$	$1$
168.96.1.gp.1	$168$	$2$	$2$	$1$
168.96.1.gw.1	$168$	$2$	$2$	$1$
168.96.1.gx.1	$168$	$2$	$2$	$1$
168.96.1.he.2	$168$	$2$	$2$	$1$
168.96.1.hf.2	$168$	$2$	$2$	$1$
168.144.8.lq.2	$168$	$3$	$3$	$8$
168.192.7.gk.1	$168$	$4$	$4$	$7$
168.384.23.he.1	$168$	$8$	$8$	$23$