Modular curve 156.16.0.a.2

• Label: 156.16.0.a.2
• Level: $156$
• Index: $16$
• Genus: $0$
• Cusps: $2$
• $\Q$-cusps: $2$

Invariants

Level:	$156$		$\SL_2$-level:	$12$
Index:	$16$		$\PSL_2$-index:	$16$
Genus:	$0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$
Cusps:	$2$ (all of which are rational)		Cusp widths	$4\cdot12$		Cusp orbits	$1^{2}$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
$\Q$-gonality:	$1$
$\overline{\Q}$-gonality:	$1$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

12B0

Level structure

$\GL_2(\Z/156\Z)$-generators:	$\begin{bmatrix}45&52\\116&151\end{bmatrix}$, $\begin{bmatrix}71&60\\44&67\end{bmatrix}$, $\begin{bmatrix}76&143\\63&77\end{bmatrix}$, $\begin{bmatrix}135&40\\82&27\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	156.32.0-156.a.2.1, 156.32.0-156.a.2.2, 156.32.0-156.a.2.3, 156.32.0-156.a.2.4, 156.32.0-156.a.2.5, 156.32.0-156.a.2.6, 156.32.0-156.a.2.7, 156.32.0-156.a.2.8, 312.32.0-156.a.2.1, 312.32.0-156.a.2.2, 312.32.0-156.a.2.3, 312.32.0-156.a.2.4, 312.32.0-156.a.2.5, 312.32.0-156.a.2.6, 312.32.0-156.a.2.7, 312.32.0-156.a.2.8
Cyclic 156-isogeny field degree:	$84$
Cyclic 156-torsion field degree:	$4032$
Full 156-torsion field degree:	$7547904$

Models

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

Rational points

This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
6.8.0.a.1	$6$	$2$	$2$	$0$	$0$

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

Covering curve	Level	Index	Degree	Genus
156.48.2.a.1	$156$	$3$	$3$	$2$
156.48.2.c.1	$156$	$3$	$3$	$2$
156.48.2.d.2	$156$	$3$	$3$	$2$
156.48.3.a.1	$156$	$3$	$3$	$3$
156.64.1.a.1	$156$	$4$	$4$	$1$
156.224.15.c.1	$156$	$14$	$14$	$15$