Modular curve 112.384.9-56.s.4.14

• Label: 112.384.9-56.s.4.14
• Level: $112$
• Index: $384$
• Genus: $9$
• Cusps: $16$
• $\Q$-cusps: $4$

Invariants

Level:	$112$		$\SL_2$-level:	$112$		Newform level:	$56$
Index:	$384$		$\PSL_2$-index:	$192$
Genus:	$9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (of which $4$ are rational)		Cusp widths	$1^{4}\cdot2^{2}\cdot7^{4}\cdot8^{2}\cdot14^{2}\cdot56^{2}$		Cusp orbits	$1^{4}\cdot2^{6}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 9$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 9$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56C9

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}7&48\\76&35\end{bmatrix}$, $\begin{bmatrix}37&66\\80&23\end{bmatrix}$, $\begin{bmatrix}69&24\\38&111\end{bmatrix}$, $\begin{bmatrix}70&43\\87&82\end{bmatrix}$, $\begin{bmatrix}75&36\\98&13\end{bmatrix}$, $\begin{bmatrix}86&57\\81&62\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.9.s.4 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 $\mathbb{P}^{ 8 }$ defined by 21 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 64 x^{11} + 192 x^{10} y + 240 x^{9} y^{2} + 48 x^{9} z^{2} + 160 x^{8} y^{3} + 144 x^{8} y z^{2} + \cdots + y^{3} z^{8} $

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

$(0:0:0:0:-1:0:0:1:1)$, $(0:0:0:0:0:1:0:0:1)$, $(1/4:-1/2:0:-1/2:-1/4:1/4:0:-1/2:1)$, $(-1/4:1/2:0:1/2:-1/4:1/4:0:-1/2:1)$

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 28.96.4.c.4 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle w$
$\displaystyle W$	$=$	$\displaystyle z$

Equation of the image curve:

$0$	$=$	$ 4X^{2}+2XY+Y^{2}+2XZ-2XW-YW-ZW $
	$=$	$ 3X^{2}Y+XY^{2}-X^{2}Z-XYZ+XZ^{2}+X^{2}W+XZW+XW^{2}+YW^{2} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.9.s.4 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle t$

Equation of the image curve:

$0$

$=$

$ 64X^{11}+192X^{10}Y+240X^{9}Y^{2}+48X^{9}Z^{2}+160X^{8}Y^{3}+144X^{8}YZ^{2}+60X^{7}Y^{4}+168X^{7}Y^{2}Z^{2}+25X^{7}Z^{4}+12X^{6}Y^{5}+96X^{6}Y^{3}Z^{2}+75X^{6}YZ^{4}+X^{5}Y^{6}+27X^{5}Y^{4}Z^{2}+83X^{5}Y^{2}Z^{4}+6X^{5}Z^{6}+3X^{4}Y^{5}Z^{2}+41X^{4}Y^{3}Z^{4}+18X^{4}YZ^{6}+8X^{3}Y^{4}Z^{4}+18X^{3}Y^{2}Z^{6}+X^{3}Z^{8}+6X^{2}Y^{3}Z^{6}+3X^{2}YZ^{8}+3XY^{2}Z^{8}+Y^{3}Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.192.5-56.bl.1.26	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$