Modular curve 136.48.0.s.1

• Label: 136.48.0.s.1
• Level: $136$
• Index: $48$
• Genus: $0$
• Cusps: $10$
• $\Q$-cusps: $0$

Invariants

Level:	$136$		$\SL_2$-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

$\GL_2(\Z/136\Z)$-generators:	$\begin{bmatrix}41&104\\108&79\end{bmatrix}$, $\begin{bmatrix}41&120\\78&117\end{bmatrix}$, $\begin{bmatrix}55&112\\16&97\end{bmatrix}$, $\begin{bmatrix}97&44\\94&49\end{bmatrix}$, $\begin{bmatrix}121&52\\104&41\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	136.96.0-136.s.1.1, 136.96.0-136.s.1.2, 136.96.0-136.s.1.3, 136.96.0-136.s.1.4, 136.96.0-136.s.1.5, 136.96.0-136.s.1.6, 136.96.0-136.s.1.7, 136.96.0-136.s.1.8, 136.96.0-136.s.1.9, 136.96.0-136.s.1.10, 136.96.0-136.s.1.11, 136.96.0-136.s.1.12, 136.96.0-136.s.1.13, 136.96.0-136.s.1.14, 136.96.0-136.s.1.15, 136.96.0-136.s.1.16
Cyclic 136-isogeny field degree:	$36$
Cyclic 136-torsion field degree:	$2304$
Full 136-torsion field degree:	$2506752$

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.e.1	$8$	$2$	$2$	$0$	$0$
136.24.0.h.2	$136$	$2$	$2$	$0$	$?$
136.24.0.l.1	$136$	$2$	$2$	$0$	$?$

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

Covering curve	Level	Index	Degree	Genus
136.96.1.g.2	$136$	$2$	$2$	$1$
136.96.1.j.2	$136$	$2$	$2$	$1$
136.96.1.w.1	$136$	$2$	$2$	$1$
136.96.1.z.1	$136$	$2$	$2$	$1$
136.96.1.bc.1	$136$	$2$	$2$	$1$
136.96.1.bd.1	$136$	$2$	$2$	$1$
136.96.1.bg.2	$136$	$2$	$2$	$1$
136.96.1.bh.2	$136$	$2$	$2$	$1$