Modular curve 112.384.11-56.fh.3.2

• Label: 112.384.11-56.fh.3.2
• Level: $112$
• Index: $384$
• Genus: $11$
• Cusps: $12$
• $\Q$-cusps: $6$

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 $6$ are rational)		Cusp widths	$2^{4}\cdot8^{2}\cdot14^{4}\cdot56^{2}$		Cusp orbits	$1^{6}\cdot2^{3}$
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:	$6$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

56N11

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}13&70\\16&99\end{bmatrix}$, $\begin{bmatrix}38&79\\101&32\end{bmatrix}$, $\begin{bmatrix}41&68\\24&53\end{bmatrix}$, $\begin{bmatrix}62&33\\23&80\end{bmatrix}$, $\begin{bmatrix}71&62\\66&19\end{bmatrix}$, $\begin{bmatrix}93&80\\76&33\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fh.3 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 $	$=$	$ y a - y b + t s $
	$=$	$y b + w r + t r$
	$=$	$x^{2} - x v + x a - x b - r s$
	$=$	$x s - v s + r a - r b + s a$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 4 x^{8} y^{4} z^{2} + 4 x^{8} y^{3} z^{3} + x^{8} y^{2} z^{4} + 160 x^{6} y^{7} z + 328 x^{6} y^{6} z^{2} + \cdots + 4 y^{2} z^{12} $

Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(1/3:0:0:0:0:0:1/3:1/3:-4/3:-1/3:1)$, $(0:0:0:0:0:0:0:-1:0:1:1)$, $(0:0:0:0:0:0:0:1:0:1:1)$, $(1/3:0:0:0:0:0:1/3:-1/3:4/3:-1/3:1)$, $(1/3:0:0:2/3:-1/3:-2/3:-1/3:0:0:1/3:1)$, $(1/3:0:0:-2/3:1/3:2/3:-1/3:0:0:1/3:1)$

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.1 :

$\displaystyle X$	$=$	$\displaystyle z$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle -r$
$\displaystyle W$	$=$	$\displaystyle -r-s$

Equation of the image curve:

$0$	$=$	$ 7X^{2}+28XY+28Y^{2}+3Z^{2}-2ZW-W^{2} $
	$=$	$ 7X^{2}Y-14XY^{2}+2XZ^{2}-5YZ^{2}-4XZW-2YZW+3YW^{2} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.192.11.fh.3 :

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

Equation of the image curve:

$0$

$=$

$ 4X^{8}Y^{4}Z^{2}+4X^{8}Y^{3}Z^{3}+X^{8}Y^{2}Z^{4}+160X^{6}Y^{7}Z+328X^{6}Y^{6}Z^{2}+320X^{6}Y^{5}Z^{3}+198X^{6}Y^{4}Z^{4}+78X^{6}Y^{3}Z^{5}+18X^{6}Y^{2}Z^{6}+2X^{6}YZ^{7}+576X^{4}Y^{10}+2400X^{4}Y^{9}Z+4636X^{4}Y^{8}Z^{2}+5748X^{4}Y^{7}Z^{3}+4983X^{4}Y^{6}Z^{4}+3086X^{4}Y^{5}Z^{5}+1381X^{4}Y^{4}Z^{6}+444X^{4}Y^{3}Z^{7}+99X^{4}Y^{2}Z^{8}+14X^{4}YZ^{9}+X^{4}Z^{10}-864X^{2}Y^{11}Z-3384X^{2}Y^{10}Z^{2}-5312X^{2}Y^{9}Z^{3}-5890X^{2}Y^{8}Z^{4}-4594X^{2}Y^{7}Z^{5}-2480X^{2}Y^{6}Z^{6}-932X^{2}Y^{5}Z^{7}-242X^{2}Y^{4}Z^{8}-42X^{2}Y^{3}Z^{9}-4X^{2}Y^{2}Z^{10}+4356Y^{12}Z^{2}+15972Y^{11}Z^{3}+29161Y^{10}Z^{4}+33352Y^{9}Z^{5}+26290Y^{8}Z^{6}+14872Y^{7}Z^{7}+6165Y^{6}Z^{8}+1868Y^{5}Z^{9}+400Y^{4}Z^{10}+56Y^{3}Z^{11}+4Y^{2}Z^{12} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.192.5-56.bl.1.13	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$