Modular curve 112.384.9-56.s.2.14

• Label: 112.384.9-56.s.2.14
• 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:	$\begin{bmatrix}25&78\\96&7\end{bmatrix}$, $\begin{bmatrix}77&44\\90&87\end{bmatrix}$, $\begin{bmatrix}93&90\\66&61\end{bmatrix}$, $\begin{bmatrix}96&71\\63&104\end{bmatrix}$, $\begin{bmatrix}101&10\\110&57\end{bmatrix}$, $\begin{bmatrix}105&68\\12&49\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.9.s.2 for the level structure with $-I$)
Cyclic 112-isogeny field degree:	$2$
Cyclic 112-torsion field degree:	$48$
Full 112-torsion field degree:	$129024$

Models

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

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - 81 x^{10} - 162 x^{9} z - 54 x^{8} y^{2} - 315 x^{8} z^{2} + 105 x^{7} y^{2} z - 378 x^{7} z^{3} + \cdots + 16 y^{4} z^{6} $

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

$(0:0:0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:1:0:0)$, $(0:-2:1:0:-2:3:-1:-1:1)$, $(0:-2:-1:0:-2:-3:-1:-1:1)$

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 28.96.4.c.2 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle -w$
$\displaystyle Z$	$=$	$\displaystyle z$
$\displaystyle W$	$=$	$\displaystyle u$

Equation of the image curve:

$0$	$=$	$ X^{2}+4XY+2Y^{2}+3XZ-YZ-3Z^{2}+XW+YW-2ZW+W^{2} $
	$=$	$ 5X^{3}+X^{2}Y-2Y^{3}+X^{2}Z-3XYZ-XZ^{2}+2YZ^{2}+X^{2}W-Y^{2}W-2XZW-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.2 :

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

Equation of the image curve:

$0$

$=$

$ -81X^{10}-162X^{9}Z-54X^{8}Y^{2}-315X^{8}Z^{2}+105X^{7}Y^{2}Z-378X^{7}Z^{3}-46X^{6}Y^{4}+219X^{6}Y^{2}Z^{2}-385X^{6}Z^{4}+151X^{5}Y^{4}Z+252X^{5}Y^{2}Z^{3}-280X^{5}Z^{5}-30X^{4}Y^{6}+270X^{4}Y^{4}Z^{2}+186X^{4}Y^{2}Z^{4}-168X^{4}Z^{6}+33X^{3}Y^{6}Z+254X^{3}Y^{4}Z^{3}+72X^{3}Y^{2}Z^{5}-64X^{3}Z^{7}-14X^{2}Y^{8}+57X^{2}Y^{6}Z^{2}+167X^{2}Y^{4}Z^{4}+24X^{2}Y^{2}Z^{6}-16X^{2}Z^{8}-7XY^{8}Z+48XY^{6}Z^{3}+48XY^{4}Z^{5}-7Y^{8}Z^{2}+24Y^{6}Z^{4}+16Y^{4}Z^{6} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.192.5-56.bl.1.28	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$