LMFDB - Modular curve 112.384.11-56.fa.1.30

Invariants

Level:	$112$	$\SL_2$-level:	$112$	Newform level:	$56$
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	$1^{2}\cdot2\cdot4\cdot7^{2}\cdot8^{2}\cdot14\cdot28\cdot56^{2}$	Cusp orbits	$1^{6}\cdot2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 11$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 11$
Rational cusps:	$6$
Rational CM points:	none

Level structure

$\GL_2(\Z/112\Z)$-generators:	$\begin{bmatrix}16&27\\29&70\end{bmatrix}$, $\begin{bmatrix}36&71\\65&98\end{bmatrix}$, $\begin{bmatrix}50&57\\3&48\end{bmatrix}$, $\begin{bmatrix}68&109\\21&100\end{bmatrix}$, $\begin{bmatrix}92&25\\31&86\end{bmatrix}$, $\begin{bmatrix}96&15\\1&54\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 56.192.11.fa.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 y + x u - x v - x s + x a - y s + w a + u s + s^{2} + s b $
	$=$	$x w + x r + y s - 2 y a - w a - u s - s^{2} + s b + a^{2} - a b$
	$=$	$x^{2} + x y + x z + x w - x v + x s + x b + y z - y v + y s + y b - z w + w a$
	$=$	$2 x^{2} - x y + 2 x z + x w - x u - x v + x s + y z + y v - y s + y a - y b + z t - z v + w a + v s + \cdots + a b$
	$=$	$\cdots$

[x*y + x*u - x*v - x*s + x*a - y*s + w*a + u*s + s^2 + s*b, x*w + x*r + y*s - 2*y*a - w*a - u*s - s^2 + s*b + a^2 - a*b, x^2 + x*y + x*z + x*w - x*v + x*s + x*b + y*z - y*v + y*s + y*b - z*w + w*a, 2*x^2 - x*y + 2*x*z + x*w - x*u - x*v + x*s + y*z + y*v - y*s + y*a - y*b + z*t - z*v + w*a + v*s - v*a + v*b - s^2 + 2*s*a - s*b - a^2 + a*b, x^2 - x*y - x*u + x*v + x*a - x*b - y*w + y*v - y*r + y*s - y*a - y*b + z*t - w*a - u*v - 2*v*s + s^2 - 2*s*a + s*b + a^2 - a*b, x^2 - 2*x*y - x*v - x*r + x*s + x*a + x*b + y^2 + y*z + y*u + y*v - y*s - z*w + z*t - z*v + w*a - u*b + v*r - v*s + s^2 + s*b, x*z - x*v - x*r + x*b + y^2 + y*w + y*u - y*s + 2*y*b - z*w + z*v + w*a - u*r + u*s - u*a - v^2 + v*r + v*s - s^2 + s*a, x^2 - x*y - x*u + x*v + x*r + x*s - x*b - y^2 - y*u + y*v + y*a - y*b + z*w + t*b - u*s + u*a - v*s - r*s + r*a, x*w - x*t + x*s + x*b + y^2 + y*z + y*u + y*s - y*a - z*w + z*t - z*v - w*s - t*b - u*s + v^2 - v*s, x*y - x*w + x*t - x*u - x*v + x*s - y*s + w*s + u*s - v*s + r*s - r*a + s^2 + s*b, x^2 + x*z - x*s - y^2 + y*w - y*t + y*u - y*v + y*s + y*a + y*b - z*t + z*v + t*s + t*b - u*a - v^2 + v*s - s^2 - s*b, x*y + x*w + x*u - x*a - y*w - y*u - y*v + y*s - w*u - u^2 - u*s + u*a - v*r + v*s - s*a - s*b, 2*x*y + x*t - x*v + x*r + 2*y^2 + y*t - y*v + y*r - y*a - t*s + v*s + r*s - r*a, x*w + x*t + x*u - x*v + x*b + y^2 + y*w + y*u + y*r + y*s - y*a + y*b + z*t + w*a - t*v + v*s, x*v - x*b - y*z + y*a + z*w - z*t + z*v - w*u - w*s + t*v + t*b - u*v - u*r - v^2 - s^2 - s*b, x*y - x*t + x*u - x*r - x*s + x*a - y*w - y*r + y*s - y*a - z*w - w*s + w*a + t*v - 2*t*s + t*a - t*b - u*v + u*s, x*y + x*u - x*s - 2*y^2 - y*t - y*u - y*v + 2*y*a - z*t + w*u - t*u + t*s + u^2 + u*v + u*r - v*r - r*s + r*a + s^2 - s*a, x^2 + 2*x*z + x*w - x*v - x*r - x*s + 2*x*a + y*u + y*v + y*s + z*t + z*s - w*s + w*a - t*b + u^2 - u*v + u*s - u*a - v*a + v*b, x^2 + x*y + x*z - x*t + x*s - y^2 + y*z - y*r + y*a - z*w + z*s + t*u - u^2 - v*s + s^2 - s*a, x*t - x*s + x*a - x*b - y^2 - y*w - 2*y*u - 2*y*s + y*a - y*b + z*w - t*u - t*a + u^2 + u*s - s^2 + s*a, x*w + x*t + x*u - x*v + x*b - y^2 - y*u - y*s + y*a + y*b - w*s + w*a - t*v + u*v + v*s - r*s + r*a, x*y + x*r + y*w + z*w - w*r - w*s + t*b - u*r - r^2 - s^2 + s*a, x^2 - x*w - x*u + x*r + x*s - x*b - y*w - y*u - y*r - y*b + z*s + w*s + t*b - u^2 - u*s + u*a - v*s, x^2 + x*w - x*u - x*v + 2*x*r + x*a + y^2 + y*z + y*t - y*u - y*s - y*a - y*b - z*v - z*s + w*r - t*r - t*a + u*v + u*r + v^2 + v*r + s*a + s*b, x*y - x*w + x*v + x*s - x*b - y*w - y*r - y*b - w*r - w*a + t^2 - t*u - t*s - u*v - u*r + u*s - v*r - v*s + r*s - r*a - s*a - s*b, x^2 + x*y + x*z - x*w - 2*x*u + x*v + x*s - x*a - x*b - y*r + y*a + z*v + w*u - w*b + t*a + t*b - v^2 - v*s + s^2 - s*a, x*z - x*u - y^2 + y*z - y*u + y*a - w*v + w*s - r*s + r*a, x^2 + x*z + x*w - x*v + x*s + y*v - y*s - y*b + z*t - z*v + z*s + w*t + w*s + t*r + t*s - v*r - v*a + v*b + s^2 + s*b, x*u - x*v - x*r + x*b + y*z + y*t - z*w + z*t - z*v + w*u - w*s + 2*w*a + t*u - t*v - t*b + u*r + u*a + v^2 + s^2 + s*a - a^2 + a*b, x^2 - x*w - x*u - x*b - 2*y^2 - y*u - y*r + y*a - y*b + z*s - w*t + w*r - u^2 + u*r - u*s + u*a - v*s, x^2 - x*y + x*z + x*v - x*b + y*a + z*v - w^2 - w*u - w*r + t*s - t*a + t*b - u*v - u*r - v^2 - s^2 - s*b, x*u - x*r - x*s + x*a + y^2 + y*u + y*s - y*a - z*w - z*s + z*a + t*v - u*v - u*b - v^2 + s^2 - 2*s*a + s*b + a^2 - a*b, x^2 - x*w - x*s + x*a - x*b - y*w - y*r - y*a - y*b + z*u + w*u + w*s - u*v + u*r + u*s - v*s + r*s - r*a + s^2 + s*b, x*w + x*s + x*b + y^2 + y*u - z*w - z*r - u*a + v*r, x*z - x*s + x*a - x*b - y*z - 2*y*w - y*u - y*r - y*s + y*a + z*w - w*v - t*u + u^2 + u*s - v*r - v*s + r*s + r*b + s^2 - s*a, 2*x*y + x*u - x*s - x*a + y*w - y*t - 2*y*v + y*r + y*s + y*b + 2*z^2 - z*t + z*r - z*b - w*s + w*b - t*u - u*s + u*b + v*a - v*b - r*s - r*b - s^2 + s*b - a*b + b^2]

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

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 $X_0(56)$ :

$\displaystyle X$	$=$	$\displaystyle w+u$
$\displaystyle Y$	$=$	$\displaystyle t-u$
$\displaystyle Z$	$=$	$\displaystyle w-t+u+v+r$
$\displaystyle W$	$=$	$\displaystyle w+u+2s-a$
$\displaystyle T$	$=$	$\displaystyle -x-z+2b$

Equation of the image curve:

$0$	$=$	$ YZ+XW+XT $
	$=$	$ X^{2}+Y^{2}-XZ-YW-2W^{2}-WT $
	$=$	$ 2X^{2}-Y^{2}+XZ+Z^{2}-YW-YT $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
112.192.5-56.bl.1.6	$112$	$2$	$2$	$5$	$?$
112.192.5-56.bl.1.31	$112$	$2$	$2$	$5$	$?$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi
Siegel

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Modular curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Galois groups
Sato-Tate groups
Abstract groups
Lattices

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Motives

Groups

Database

Properties

Related objects

Downloads

Learn more