Modular curve 112.384.11-56.fe.2.27

• Label: 112.384.11-56.fe.2.27
• Level: $112$
• Index: $384$
• Genus: $11$
• Cusps: $12$
• $\Q$-cusps: $8$

Invariants

Level:	$112$		$\SL_2$-level:	$112$		Newform level:	$56$
Index:	$384$		$\PSL_2$-index:	$192$
Genus:	$11 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $8$ are rational)		Cusp widths	$1^{2}\cdot2\cdot4\cdot7^{2}\cdot8^{2}\cdot14\cdot28\cdot56^{2}$		Cusp orbits	$1^{8}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 11$
Rational cusps:	$8$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56P11

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}2&45\\91&12\end{bmatrix}$, $\begin{bmatrix}25&32\\44&13\end{bmatrix}$, $\begin{bmatrix}25&70\\90&61\end{bmatrix}$, $\begin{bmatrix}78&33\\77&90\end{bmatrix}$, $\begin{bmatrix}83&6\\106&95\end{bmatrix}$, $\begin{bmatrix}97&54\\44&51\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fe.2 for the level structure with $-I$)
Cyclic 112-isogeny field degree:	$2$
Cyclic 112-torsion field degree:	$24$
Full 112-torsion field degree:	$129024$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

$ 0 $	$=$	$ x y + x z - x w + x t + x v + x r - x s + y z + y t + y v + y r - y b - t r - v^{2} - v a - s^{2} + s b $
	$=$	$x y - x z - x w - x t - 2 x v + x r - x a + x b - y^{2} - y t + y v + y r + z t - u v - v s$
	$=$	$2 x y - x z - x w - x u + y^{2} - y z - y t + y r + z t + w v - v r$
	$=$	$2 x t - x u + x v - x b + 2 y t + y v + y r - z t + w v - v r - r s$
	$=$	$\cdots$

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

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

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 $X_0(56)$ :

$\displaystyle X$	$=$	$\displaystyle -y+z$
$\displaystyle Y$	$=$	$\displaystyle -x-y-z+w$
$\displaystyle Z$	$=$	$\displaystyle -y+z+u+r$
$\displaystyle W$	$=$	$\displaystyle -y+z+t+v$
$\displaystyle T$	$=$	$\displaystyle a-b$

Equation of the image curve:

$0$	$=$	$ YZ+XW+XT $
	$=$	$ X^{2}+Y^{2}-XZ-YW-2W^{2}-WT $
	$=$	$ 2X^{2}-Y^{2}+XZ+Z^{2}-YW-YT $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(7)$	$7$	$48$	$24$	$0$	$0$
16.48.0-8.bb.1.8	$16$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.bb.1.8	$16$	$8$	$8$	$0$	$0$
112.192.5-56.bl.1.21	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$