LMFDB - Modular curve 84.384.11-42.l.1.16

Invariants

Level:	$84$	$\SL_2$-level:	$84$	Newform level:	$1764$
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	$2^{3}\cdot6^{3}\cdot14^{3}\cdot42^{3}$	Cusp orbits	$1^{4}\cdot2^{4}$
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:	$4$
Rational CM points:	none

Level structure

$\GL_2(\Z/84\Z)$-generators:	$\begin{bmatrix}1&42\\28&47\end{bmatrix}$, $\begin{bmatrix}10&15\\7&68\end{bmatrix}$, $\begin{bmatrix}22&47\\63&2\end{bmatrix}$, $\begin{bmatrix}46&21\\21&34\end{bmatrix}$, $\begin{bmatrix}51&68\\28&65\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 42.192.11.l.1 for the level structure with $-I$)
Cyclic 84-isogeny field degree:	$2$
Cyclic 84-torsion field degree:	$48$
Full 84-torsion field degree:	$24192$

Models

Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations

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

[x*y + y*z - z*u, x*y + y*r - y*s + y*a - u*v, x*t - y*v - y*s - z*t - w*t + w*v + w*s, x*z - x*w - x*t - x*v + x*r + x*a + y*z + y*s - z*w - w*s, x*z + x*w + 2*x*r + x*a - y*v + y*r - y*s + u*v - u*r + u*s, x*w - y^2 + y*w + y*t - y*v - y*s + z*w - w*u + w*s - w*a - t*u - u*r + u*s, x*y - y*z + y*t - y*s + y*b - w*u - w*b - t*u - 2*u*r + u*s, x^2 - x*y + x*z + x*u + x*s + 2*x*a + x*b - y*t + y*a - w*a + t*u, 2*x*u - y*z + y*u + y*v - y*s + y*a - z*u - w*u - w*b - u*r + u*b, x^2 + x*y - x*t + 2*x*v + x*s + x*a + x*b - w*v + w*r - w*s + u*v - u*r + u*s, x*z + x*t - x*v + x*r + x*a + y*t + z*t - z*u - w*u - w*v - w*s - w*a - u*r, x*u - x*v + x*a + y*b + z^2 - z*u + z*s + w*v + w*a - w*b - u*r, x*w + x*r - x*s + y*r + z*t + z*u + z*v + z*s + w*t + w*u + w*a, x*z - x*v - z*v + z*r - z*s + z*a, x^2 - x*z + 2*x*r - x*s + y*r + z*v - w*s + w*a, x*z - x*v - y*v + y*r - y*s - z^2 - z*w - z*v - z*r - z*s, x*y - x*w - x*t - z*v - z*r - w*r + w*s - w*a + u*r, x*w + x*v + x*r - y*z + y*w + y*t - y*a - z*w - z*v - z*r - w^2 - w*t - w*v - w*s + u*v, x^2 - x*z + x*w + x*t + x*v + x*a + x*b - y*v + z*w + z*u + z*r - z*a - z*b - w*v + w*r - w*a + w*b, x*y + x*w + x*t + x*s + y*t + y*r + y*s + z*w + z*t - z*v - z*s + w*t - w*v - w*r - w*a + w*b + t*v - t*r + t*b + u*r + v*s + r^2 - r*s + r*a + s*b, x*y - x*t + x*u - x*v + y*b + z*t - z*b + w*t - w*b + t*r - u*v - v^2 + v*r - v*s - v*a - r^2 + 2*r*s - r*b - s^2 - a*b, x*y - x*w - 2*x*t + x*v - x*a - y*z + y*v - y*r + z*u + z*v + z*a - w*b + t*r - t*b - u*v - v*r + v*s - 2*v*a - r^2 - r*b - s*a - a*b, x*t - x*v + y*z - y*r + y*s - z^2 - z*w - z*t - z*r - z*s - z*a + w*v + t*r - t*s + t*a + t*b + u*r + v*r - v*b + r^2 - r*s + 2*r*a - s*a, x*w - x*t + 2*x*v + x*s - y*z + y*r + z*t - z*v - z*s + z*b + w*t + w*u + w*a + t*v + t*s - t*a + u*v - v*r + r*s - r*a + r*b - s^2 + a*b, x^2 - x*z + x*w + 2*x*u + x*r + 2*x*s - 2*x*b + y*z + y*t - y*u - y*v - y*b - z*w + z*t + z*u + z*b + w*u - w*a + w*b + t*r + u*s - u*b + r*s - s^2, x^2 + x*y + x*w - x*t + x*s - x*b - y^2 - y*z - y*w + y*t - y*r - z^2 + z*w + z*r + z*s + 2*w^2 - w*t + w*u - w*v + w*r - w*s - w*a - t*u - t*v - u*r + u*s - v^2 + v*r - v*s + v*a + r^2 + s*a, x*y - x*z + x*t + x*u - x*v - x*a - x*b - 3*y^2 - y*z - y*w - y*r + y*s - y*b - z^2 + z*w - z*t + z*u + z*v - 2*z*s - z*a + z*b - w^2 + 3*w*u + w*v - w*a + w*b - t*u + v^2 + v*a + r^2 - r*s + r*b + a*b, x^2 + 2*x*y - x*t - 2*x*u - 3*x*v + x*a + y^2 - 2*y*z - y*t - 2*y*u - y*v - y*r + y*a - 2*y*b - z^2 - z*t - z*u + z*v - z*r - z*s - z*a - w^2 + 2*w*u + 2*w*v + w*r - w*b + t*v - t*r + t*s - t*a + 2*u*v - u*r - u*s + 3*u*b, x*y - 2*x*w + x*t - 2*x*u + x*v + 2*x*r - x*b - 2*y^2 - y*z + y*w - y*t + 4*y*u + y*v + y*r - y*a + 2*y*b - z^2 - z*w + z*u - z*v + 2*z*r + z*s - z*a - z*b + w^2 - w*v - w*r - w*s + w*a + t*u - u*b - v^2 - v*a - v*b, x*y - x*z - 2*x*w + x*t + x*u - x*v - x*r + y^2 + 2*y*w - y*t - 4*y*u - y*r - y*b + z^2 - 2*z*w + z*t + z*u + z*r + 2*w^2 + w*t - w*u - w*v + w*r + t*u - t*r - t*b + 2*u*v - u*s + u*b + v*s - v*b + r^2 - 2*r*s + 2*r*a + s^2 - s*a, x^2 + x*y + 3*x*w - x*u + x*r + 2*x*s + x*a - 2*y^2 - y*w - 2*y*t + y*v - y*r + y*b + z^2 - 3*z*w + z*t + z*u - z*s + z*a + z*b + 3*w^2 - w*u - w*b + t*u - 2*t*v + t*r - t*b + u*v + v*s + r^2 - r*s + r*a + s*b, x*w - x*t - x*u - x*v - x*s + y^2 + y*w + 2*y*t + y*u - y*s + y*a - y*b + z*w - z*u + z*v + z*a - z*b - 2*w^2 + 3*w*u - w*a - w*b - t*u + t*v - t*r + t*b - 5*u^2 - u*r + u*s + u*a + u*b - v^2 + 2*v*r - v*s + 2*r*a - r*b - s*a - s*b + a^2 - a*b, x^2 - 2*x*y - x*z - 2*x*w + x*t + x*r + x*s - x*b - 3*y^2 + 2*y*z + y*t + y*v - y*s + 2*y*b + z^2 - 2*z*w + z*t - z*u - 2*z*v + 2*z*r + z*s + z*a - z*b + 3*w^2 - w*t + 2*w*u - w*v - w*r + w*s - w*a - w*b - t*v - u*v - v^2 + v*s - v*a + r^2 - r*s + r*a - r*b + s*b - a*b, x*z + x*w - 2*x*u - x*v + x*r - y^2 + y*z - 3*y*w + y*t - y*u + 2*y*v + y*a - y*b - z^2 + 2*z*w + z*u - 2*z*v - 2*z*r - z*s - z*b - w^2 - w*t - w*u + w*v + w*r - w*s + 2*w*a - 2*w*b + t*r - t*a + t*b + u*v - 2*u*s + 4*u*b + v^2 + v*s - v*a + v*b - r^2 + r*s - r*b - s^2 + s*b - a*b, x^2 + x*y - x*z - x*w + x*t - x*u - 2*x*v + x*s - x*b + y^2 + y*z + 2*y*w - y*r - y*b + z*w + 3*z*u + z*v - z*r - 2*z*s + 2*z*b + 2*w^2 - w*t + 2*w*u - w*r - 2*w*s - w*a - w*b + t*u + t*v - t*r + t*s - t*a - t*b + u*v - u*r - u*s - 2*v*r + v*s - v*b - r*a + a*b, 3*x^2 + x*y + 4*x*z + x*w - 2*x*a - 3*x*b + y^2 + y*w - y*t + y*u + y*r - y*b + 3*z^2 - z*t - z*r + z*s - 3*z*a - z*b - 2*w^2 + w*t - 2*w*u + w*v - w*r + w*s + w*a + w*b + t*v - t*r + t*b + u*v + u*r + u*b + v*r + r*a]

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

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 $X_0(42)$ :

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

Equation of the image curve:

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

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
84.48.0-42.c.1.1	$84$	$8$	$8$	$0$	$?$
84.192.5-42.a.1.8	$84$	$2$	$2$	$5$	$?$
84.192.5-42.a.1.47	$84$	$2$	$2$	$5$	$?$

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