Modular curve 168.48.0.cy.1

• Label: 168.48.0.cy.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	$2^{5}$
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:	$\begin{bmatrix}13&48\\46&1\end{bmatrix}$, $\begin{bmatrix}45&64\\20&109\end{bmatrix}$, $\begin{bmatrix}45&160\\70&61\end{bmatrix}$, $\begin{bmatrix}49&152\\162&43\end{bmatrix}$, $\begin{bmatrix}99&56\\148&115\end{bmatrix}$, $\begin{bmatrix}149&32\\74&135\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	168.96.0-168.cy.1.1, 168.96.0-168.cy.1.2, 168.96.0-168.cy.1.3, 168.96.0-168.cy.1.4, 168.96.0-168.cy.1.5, 168.96.0-168.cy.1.6, 168.96.0-168.cy.1.7, 168.96.0-168.cy.1.8, 168.96.0-168.cy.1.9, 168.96.0-168.cy.1.10, 168.96.0-168.cy.1.11, 168.96.0-168.cy.1.12, 168.96.0-168.cy.1.13, 168.96.0-168.cy.1.14, 168.96.0-168.cy.1.15, 168.96.0-168.cy.1.16, 168.96.0-168.cy.1.17, 168.96.0-168.cy.1.18, 168.96.0-168.cy.1.19, 168.96.0-168.cy.1.20, 168.96.0-168.cy.1.21, 168.96.0-168.cy.1.22, 168.96.0-168.cy.1.23, 168.96.0-168.cy.1.24
Cyclic 168-isogeny field degree:	$32$
Cyclic 168-torsion field degree:	$1536$
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.i.1	$8$	$2$	$2$	$0$	$0$
168.24.0.u.1	$168$	$2$	$2$	$0$	$?$
168.24.0.ed.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.bu.1	$168$	$2$	$2$	$1$
168.96.1.fp.1	$168$	$2$	$2$	$1$
168.96.1.ky.1	$168$	$2$	$2$	$1$
168.96.1.lc.1	$168$	$2$	$2$	$1$
168.96.1.ou.2	$168$	$2$	$2$	$1$
168.96.1.oy.2	$168$	$2$	$2$	$1$
168.96.1.pu.2	$168$	$2$	$2$	$1$
168.96.1.qc.2	$168$	$2$	$2$	$1$
168.144.8.pk.2	$168$	$3$	$3$	$8$
168.192.7.js.1	$168$	$4$	$4$	$7$
168.384.23.jw.1	$168$	$8$	$8$	$23$