LMFDB - Modular curve 84.384.11-42.o.1.23

Invariants

Level:	$84$	$\SL_2$-level:	$84$	Newform level:	$84$
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	$2^{3}\cdot6^{3}\cdot14^{3}\cdot42^{3}$	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

$\GL_2(\Z/84\Z)$-generators:	$\begin{bmatrix}16&9\\1&38\end{bmatrix}$, $\begin{bmatrix}29&60\\32&43\end{bmatrix}$, $\begin{bmatrix}33&34\\56&25\end{bmatrix}$, $\begin{bmatrix}73&48\\70&65\end{bmatrix}$, $\begin{bmatrix}74&69\\5&82\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 42.192.11.o.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 v + z v + w a $
	$=$	$y t + y r + w t + w r - u v$
	$=$	$x v + x a + z a - w v - w a + u v$
	$=$	$x z + x t - x r + y z + y t - z r$
	$=$	$\cdots$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ - 49 x^{11} z^{4} - 1113 x^{10} y^{2} z^{3} - 1029 x^{10} y z^{4} - 637 x^{10} z^{5} - 8352 x^{9} y^{4} z^{2} + \cdots - 49 z^{15} $

[-49*x^11*z^4 - 1113*x^10*y^2*z^3 - 1029*x^10*y*z^4 - 637*x^10*z^5 - 8352*x^9*y^4*z^2 - 15057*x^9*y^3*z^3 - 17661*x^9*y^2*z^4 - 11613*x^9*y*z^5 - 3430*x^9*z^6 - 22761*x^8*y^6*z - 59049*x^8*y^5*z^2 - 110277*x^8*y^4*z^3 - 140994*x^8*y^3*z^4 - 109725*x^8*y^2*z^5 - 50715*x^8*y*z^6 - 10192*x^8*z^7 - 18225*x^7*y^8 - 61965*x^7*y^7*z - 207765*x^7*y^6*z^2 - 435996*x^7*y^5*z^3 - 580203*x^7*y^4*z^4 - 525987*x^7*y^3*z^5 - 321678*x^7*y^2*z^6 - 113337*x^7*y*z^7 - 18963*x^7*z^8 - 91125*x^6*y^8*z - 333639*x^6*y^7*z^2 - 781731*x^6*y^6*z^3 - 1240218*x^6*y^5*z^4 - 1360440*x^6*y^4*z^5 - 983115*x^6*y^3*z^6 - 470904*x^6*y^2*z^7 - 150528*x^6*y*z^8 - 23765*x^6*z^9 - 171315*x^5*y^8*z^2 - 585468*x^5*y^7*z^3 - 1236708*x^5*y^6*z^4 - 1637820*x^5*y^5*z^5 - 1393983*x^5*y^4*z^6 - 855162*x^5*y^3*z^7 - 403410*x^5*y^2*z^8 - 129948*x^5*y*z^9 - 21021*x^5*z^10 - 172044*x^4*y^8*z^3 - 417393*x^4*y^7*z^4 - 557199*x^4*y^6*z^5 - 590706*x^4*y^5*z^6 - 542142*x^4*y^4*z^7 - 376425*x^4*y^3*z^8 - 205674*x^4*y^2*z^9 - 75117*x^4*y*z^10 - 13475*x^4*z^11 - 37665*x^3*y^8*z^4 + 55323*x^3*y^7*z^5 + 138915*x^3*y^6*z^6 + 98874*x^3*y^5*z^7 - 20133*x^3*y^4*z^8 - 75474*x^3*y^3*z^9 - 51156*x^3*y^2*z^10 - 27489*x^3*y*z^11 - 6272*x^3*z^12 - 34101*x^2*y^8*z^5 - 1944*x^2*y^7*z^6 + 36693*x^2*y^6*z^7 - 5103*x^2*y^5*z^8 - 2664*x^2*y^4*z^9 + 21420*x^2*y^3*z^10 - 2121*x^2*y^2*z^11 - 5145*x^2*y*z^12 - 2058*x^2*z^13 - 81*x*y^8*z^6 + 2430*x*y^7*z^7 - 11664*x*y^6*z^8 + 9882*x*y^5*z^9 + 2259*x*y^4*z^10 - 6552*x*y^3*z^11 + 3633*x*y^2*z^12 + 147*x*y*z^13 - 441*x*z^14 - 441*y^4*z^11 + 882*y^3*z^12 - 735*y^2*z^13 + 294*y*z^14 - 49*z^15]

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
$(0:0:0:0:0:0:0:0:1:0:0)$, $(0:0:0:0:0:0:0:0:0:0:1)$, $(0:0:0:0:0:0:1:1:0:0:0)$, $(1/6:-1/3:0:1/6:1/2:0:1/6:-1/2:1/2:-1/6:1)$, $(-1/6:1/3:0:-1/6:0:0:1/3:0:1/2:-1/3:1)$, $(0:0:0:0:1:0:-1:2:0:1:1)$

Maps to other modular curves

Map of degree 3 from the canonical model of this modular curve to the canonical model of the modular curve 42.64.3.e.1 :

$\displaystyle X$	$=$	$\displaystyle -21x+21w$
$\displaystyle Y$	$=$	$\displaystyle 3x+3y+3z-3u+6v-6r+6s-6a+3b$
$\displaystyle Z$	$=$	$\displaystyle -5x+2y-5z-2u-3v-4r-3s+3a+2b$

Equation of the image curve:

$0$

$=$

$ 3X^{4}-2X^{2}Y^{2}-8X^{2}YZ+2Y^{3}Z+6X^{2}Z^{2}+Y^{2}Z^{2}-6YZ^{3} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 42.192.11.o.1 :

$\displaystyle X$	$=$	$\displaystyle v$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{3}b$
$\displaystyle Z$	$=$	$\displaystyle a$

Equation of the image curve:

$0$

$=$

$ -18225X^{7}Y^{8}-22761X^{8}Y^{6}Z-61965X^{7}Y^{7}Z-91125X^{6}Y^{8}Z-8352X^{9}Y^{4}Z^{2}-59049X^{8}Y^{5}Z^{2}-207765X^{7}Y^{6}Z^{2}-333639X^{6}Y^{7}Z^{2}-171315X^{5}Y^{8}Z^{2}-1113X^{10}Y^{2}Z^{3}-15057X^{9}Y^{3}Z^{3}-110277X^{8}Y^{4}Z^{3}-435996X^{7}Y^{5}Z^{3}-781731X^{6}Y^{6}Z^{3}-585468X^{5}Y^{7}Z^{3}-172044X^{4}Y^{8}Z^{3}-49X^{11}Z^{4}-1029X^{10}YZ^{4}-17661X^{9}Y^{2}Z^{4}-140994X^{8}Y^{3}Z^{4}-580203X^{7}Y^{4}Z^{4}-1240218X^{6}Y^{5}Z^{4}-1236708X^{5}Y^{6}Z^{4}-417393X^{4}Y^{7}Z^{4}-37665X^{3}Y^{8}Z^{4}-637X^{10}Z^{5}-11613X^{9}YZ^{5}-109725X^{8}Y^{2}Z^{5}-525987X^{7}Y^{3}Z^{5}-1360440X^{6}Y^{4}Z^{5}-1637820X^{5}Y^{5}Z^{5}-557199X^{4}Y^{6}Z^{5}+55323X^{3}Y^{7}Z^{5}-34101X^{2}Y^{8}Z^{5}-3430X^{9}Z^{6}-50715X^{8}YZ^{6}-321678X^{7}Y^{2}Z^{6}-983115X^{6}Y^{3}Z^{6}-1393983X^{5}Y^{4}Z^{6}-590706X^{4}Y^{5}Z^{6}+138915X^{3}Y^{6}Z^{6}-1944X^{2}Y^{7}Z^{6}-81XY^{8}Z^{6}-10192X^{8}Z^{7}-113337X^{7}YZ^{7}-470904X^{6}Y^{2}Z^{7}-855162X^{5}Y^{3}Z^{7}-542142X^{4}Y^{4}Z^{7}+98874X^{3}Y^{5}Z^{7}+36693X^{2}Y^{6}Z^{7}+2430XY^{7}Z^{7}-18963X^{7}Z^{8}-150528X^{6}YZ^{8}-403410X^{5}Y^{2}Z^{8}-376425X^{4}Y^{3}Z^{8}-20133X^{3}Y^{4}Z^{8}-5103X^{2}Y^{5}Z^{8}-11664XY^{6}Z^{8}-23765X^{6}Z^{9}-129948X^{5}YZ^{9}-205674X^{4}Y^{2}Z^{9}-75474X^{3}Y^{3}Z^{9}-2664X^{2}Y^{4}Z^{9}+9882XY^{5}Z^{9}-21021X^{5}Z^{10}-75117X^{4}YZ^{10}-51156X^{3}Y^{2}Z^{10}+21420X^{2}Y^{3}Z^{10}+2259XY^{4}Z^{10}-13475X^{4}Z^{11}-27489X^{3}YZ^{11}-2121X^{2}Y^{2}Z^{11}-6552XY^{3}Z^{11}-441Y^{4}Z^{11}-6272X^{3}Z^{12}-5145X^{2}YZ^{12}+3633XY^{2}Z^{12}+882Y^{3}Z^{12}-2058X^{2}Z^{13}+147XYZ^{13}-735Y^{2}Z^{13}-441XZ^{14}+294YZ^{14}-49Z^{15} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
84.192.5-42.a.1.45	$84$	$2$	$2$	$5$	$?$
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