Modular curve 112.384.11-56.dz.1.1

• Label: 112.384.11-56.dz.1.1
• Level: $112$
• Index: $384$
• Genus: $11$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$112$		$\SL_2$-level:	$112$		Newform level:	$784$
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 $4$ are rational)		Cusp widths	$2^{4}\cdot8^{2}\cdot14^{4}\cdot56^{2}$		Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 11$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56M11

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}41&0\\6&45\end{bmatrix}$, $\begin{bmatrix}47&56\\5&67\end{bmatrix}$, $\begin{bmatrix}81&56\\106&31\end{bmatrix}$, $\begin{bmatrix}89&0\\54&17\end{bmatrix}$, $\begin{bmatrix}101&56\\104&23\end{bmatrix}$, $\begin{bmatrix}111&0\\28&85\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.dz.1 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 $\mathbb{P}^{ 10 }$ defined by 36 equations

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - 614656 x^{8} y^{10} + 614656 x^{8} y^{8} z^{2} - 98784 x^{6} y^{12} + 323792 x^{6} y^{10} z^{2} + \cdots + 4 y^{8} z^{10} $

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

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

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 28.96.5.l.1 :

$\displaystyle X$	$=$	$\displaystyle -x$
$\displaystyle Y$	$=$	$\displaystyle -y$
$\displaystyle Z$	$=$	$\displaystyle -t$
$\displaystyle W$	$=$	$\displaystyle t-r$
$\displaystyle T$	$=$	$\displaystyle a$

Equation of the image curve:

$0$	$=$	$ XZ-XW+XT+YT $
	$=$	$ 7X^{2}+ZW $
	$=$	$ 14XY+7Y^{2}-Z^{2}-6ZW-W^{2}+T^{2} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.11.dz.1 :

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle 28x$
$\displaystyle Z$	$=$	$\displaystyle 28z$

Equation of the image curve:

$0$

$=$

$ -614656X^{8}Y^{10}+614656X^{8}Y^{8}Z^{2}-98784X^{6}Y^{12}+323792X^{6}Y^{10}Z^{2}-27440X^{6}Y^{8}Z^{4}-170128X^{6}Y^{6}Z^{6}-27440X^{6}Y^{4}Z^{8}-10241X^{4}Y^{14}+32193X^{4}Y^{12}Z^{2}-11858X^{4}Y^{10}Z^{4}-34202X^{4}Y^{8}Z^{6}+3675X^{4}Y^{6}Z^{8}+11221X^{4}Y^{4}Z^{10}+2744X^{4}Y^{2}Z^{12}+196X^{4}Z^{14}-504X^{2}Y^{16}+1274X^{2}Y^{14}Z^{2}-133X^{2}Y^{12}Z^{4}-1561X^{2}Y^{10}Z^{6}-63X^{2}Y^{8}Z^{8}+679X^{2}Y^{6}Z^{10}+280X^{2}Y^{4}Z^{12}+28X^{2}Y^{2}Z^{14}-16Y^{18}-8Y^{16}Z^{2}-Y^{14}Z^{4}+13Y^{12}Z^{6}+8Y^{10}Z^{8}+4Y^{8}Z^{10} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.48.0-56.bj.1.6	$112$	$8$	$8$	$0$	$?$
112.192.5-56.bl.1.11	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$