Modular curve 112.384.11-56.cf.1.8

• Label: 112.384.11-56.cf.1.8
• Level: $112$
• Index: $384$
• Genus: $11$
• Cusps: $12$
• $\Q$-cusps: $8$

Invariants

Level:	$112$		$\SL_2$-level:	$112$		Newform level:	$112$
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	$2^{4}\cdot8^{2}\cdot14^{4}\cdot56^{2}$		Cusp orbits	$1^{8}\cdot2^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 11$
Rational cusps:	$8$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56M11

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}9&0\\0&5\end{bmatrix}$, $\begin{bmatrix}19&56\\14&107\end{bmatrix}$, $\begin{bmatrix}31&56\\32&23\end{bmatrix}$, $\begin{bmatrix}41&0\\4&89\end{bmatrix}$, $\begin{bmatrix}43&0\\99&65\end{bmatrix}$, $\begin{bmatrix}101&56\\67&71\end{bmatrix}$, $\begin{bmatrix}103&0\\9&53\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.cf.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 u - z u - v s - v b $
	$=$	$t r - t a - v s - v b$
	$=$	$2 x u - r b - s a$
	$=$	$x^{2} - w^{2} - r^{2} + r a + s^{2}$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 16 x^{8} y^{6} z^{2} - 48 x^{6} y^{8} z^{2} - 112 x^{6} y^{6} z^{4} - 16 x^{6} y^{4} z^{6} + \cdots + 36 y^{6} z^{10} $

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

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

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 28.96.5.h.1 :

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

Equation of the image curve:

$0$	$=$	$ Y^{2}-XZ $
	$=$	$ XY-ZW-YT+WT $
	$=$	$ X^{2}+3Y^{2}+3XZ-2YW+W^{2}+2ZT-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.cf.1 :

$\displaystyle X$	$=$	$\displaystyle b$
$\displaystyle Y$	$=$	$\displaystyle 2x$
$\displaystyle Z$	$=$	$\displaystyle 2y$

Equation of the image curve:

$0$

$=$

$ 16X^{8}Y^{6}Z^{2}-48X^{6}Y^{8}Z^{2}-112X^{6}Y^{6}Z^{4}-16X^{6}Y^{4}Z^{6}+16X^{4}Y^{12}+200X^{4}Y^{10}Z^{2}+449X^{4}Y^{8}Z^{4}+486X^{4}Y^{6}Z^{6}+147X^{4}Y^{4}Z^{8}+22X^{4}Y^{2}Z^{10}+X^{4}Z^{12}-16X^{2}Y^{14}-168X^{2}Y^{12}Z^{2}-529X^{2}Y^{10}Z^{4}-726X^{2}Y^{8}Z^{6}-539X^{2}Y^{6}Z^{8}-142X^{2}Y^{4}Z^{10}-9X^{2}Y^{2}Z^{12}+64Y^{14}Z^{2}+160Y^{12}Z^{4}+196Y^{10}Z^{6}+120Y^{8}Z^{8}+36Y^{6}Z^{10} $

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.q.1.2	$16$	$8$	$8$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.q.1.2	$16$	$8$	$8$	$0$	$0$
112.192.5-56.bl.1.5	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$