Modular curve 132.288.9-44.h.1.8

• Label: 132.288.9-44.h.1.8
• Level: $132$
• Index: $288$
• Genus: $9$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$132$		$\SL_2$-level:	$44$		Newform level:	$176$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps:	$8$ (of which $4$ are rational)		Cusp widths	$2^{2}\cdot4^{2}\cdot22^{2}\cdot44^{2}$		Cusp orbits	$1^{4}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 9$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 9$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

44B9

Level structure

$\GL_2(\Z/132\Z)$-generators:	$\begin{bmatrix}27&44\\116&15\end{bmatrix}$, $\begin{bmatrix}67&88\\97&117\end{bmatrix}$, $\begin{bmatrix}73&88\\116&69\end{bmatrix}$, $\begin{bmatrix}91&0\\122&7\end{bmatrix}$, $\begin{bmatrix}105&44\\17&131\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 44.144.9.h.1 for the level structure with $-I$)
Cyclic 132-isogeny field degree:	$4$
Cyclic 132-torsion field degree:	$160$
Full 132-torsion field degree:	$211200$

Models

Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations

$ 0 $	$=$	$ y r - 2 z w - z v + r s $
	$=$	$x w - x t - x u - y z - z s - v r$
	$=$	$x z + x w - x u + y w + z r - u r$
	$=$	$x z - x w + x t + y w + z r + t r$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 6125 x^{15} - 3675 x^{14} y + 2485 x^{13} y^{2} - 386 x^{13} z^{2} - 1099 x^{12} y^{3} + \cdots - y^{3} z^{12} $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(0:0:-1/2:-1/2:1/2:-1:1:0:0)$, $(0:0:1/2:-1/2:-1:1/2:1:0:0)$, $(0:0:1/2:-1/2:0:-1/2:1:0:0)$, $(0:0:-1/2:-1/2:-1/2:0:1:0:0)$

Maps to other modular curves

Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 44.72.4.a.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle z$
$\displaystyle Z$	$=$	$\displaystyle -r$
$\displaystyle W$	$=$	$\displaystyle z+t-u$

Equation of the image curve:

$0$	$=$	$ 11X^{2}+4Y^{2}+2XZ+3Z^{2}-W^{2} $
	$=$	$ X^{3}+XY^{2}-3X^{2}Z-Y^{2}Z-XZ^{2}-Z^{3} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 44.144.9.h.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle s$
$\displaystyle Z$	$=$	$\displaystyle u$

Equation of the image curve:

$0$

$=$

$ 6125X^{15}-3675X^{14}Y+2485X^{13}Y^{2}-1099X^{12}Y^{3}+335X^{11}Y^{4}-89X^{10}Y^{5}+15X^{9}Y^{6}-X^{8}Y^{7}-386X^{13}Z^{2}+2902X^{12}YZ^{2}-4473X^{11}Y^{2}Z^{2}+733X^{10}Y^{3}Z^{2}-828X^{9}Y^{4}Z^{2}+32X^{8}Y^{5}Z^{2}-25X^{7}Y^{6}Z^{2}-3X^{6}Y^{7}Z^{2}+491X^{11}Z^{4}-4033X^{10}YZ^{4}-1541X^{9}Y^{2}Z^{4}-2781X^{8}Y^{3}Z^{4}-307X^{7}Y^{4}Z^{4}-239X^{6}Y^{5}Z^{4}-35X^{5}Y^{6}Z^{4}-3X^{4}Y^{7}Z^{4}-1676X^{9}Z^{6}-36X^{8}YZ^{6}-1175X^{7}Y^{2}Z^{6}-501X^{6}Y^{3}Z^{6}-482X^{5}Y^{4}Z^{6}-86X^{4}Y^{5}Z^{6}-11X^{3}Y^{6}Z^{6}-X^{2}Y^{7}Z^{6}+835X^{7}Z^{8}+507X^{6}YZ^{8}+763X^{5}Y^{2}Z^{8}+327X^{4}Y^{3}Z^{8}+114X^{3}Y^{4}Z^{8}+14X^{2}Y^{5}Z^{8}-146X^{5}Z^{10}-146X^{4}YZ^{10}-78X^{3}Y^{2}Z^{10}-14X^{2}Y^{3}Z^{10}+5X^{3}Z^{12}+X^{2}YZ^{12}-5XY^{2}Z^{12}-Y^{3}Z^{12} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
12.24.0-4.d.1.2	$12$	$12$	$12$	$0$	$0$
$X_0(11)$	$11$	$24$	$12$	$1$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
12.24.0-4.d.1.2	$12$	$12$	$12$	$0$	$0$
132.144.4-44.a.1.2	$132$	$2$	$2$	$4$	$?$
132.144.4-44.a.1.8	$132$	$2$	$2$	$4$	$?$