Modular curve 136.48.0.v.1

• Label: 136.48.0.v.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	$2^{3}\cdot4$
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

$\GL_2(\Z/136\Z)$-generators:	$\begin{bmatrix}1&120\\42&33\end{bmatrix}$, $\begin{bmatrix}41&116\\134&49\end{bmatrix}$, $\begin{bmatrix}59&40\\90&133\end{bmatrix}$, $\begin{bmatrix}61&12\\50&25\end{bmatrix}$, $\begin{bmatrix}75&68\\130&87\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	136.96.0-136.v.1.1, 136.96.0-136.v.1.2, 136.96.0-136.v.1.3, 136.96.0-136.v.1.4, 136.96.0-136.v.1.5, 136.96.0-136.v.1.6, 136.96.0-136.v.1.7, 136.96.0-136.v.1.8, 136.96.0-136.v.1.9, 136.96.0-136.v.1.10, 136.96.0-136.v.1.11, 136.96.0-136.v.1.12, 136.96.0-136.v.1.13, 136.96.0-136.v.1.14, 136.96.0-136.v.1.15, 136.96.0-136.v.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.d.1	$8$	$2$	$2$	$0$	$0$
136.24.0.i.1	$136$	$2$	$2$	$0$	$?$
136.24.0.m.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.b.1	$136$	$2$	$2$	$1$
136.96.1.c.1	$136$	$2$	$2$	$1$
136.96.1.s.1	$136$	$2$	$2$	$1$
136.96.1.t.1	$136$	$2$	$2$	$1$
136.96.1.bi.1	$136$	$2$	$2$	$1$
136.96.1.bj.1	$136$	$2$	$2$	$1$
136.96.1.bm.1	$136$	$2$	$2$	$1$
136.96.1.bn.1	$136$	$2$	$2$	$1$