LMFDB - Modular curve 60.120.8.e.1

Invariants

Level:	$60$	$\SL_2$-level:	$20$	Newform level:	$3600$
Index:	$120$	$\PSL_2$-index:	$120$
Genus:	$8 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (none of which are rational)	Cusp widths	$20^{6}$	Cusp orbits	$2^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$3 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	20A8
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	60.120.8.37

Level structure

$\GL_2(\Z/60\Z)$-generators:	$\begin{bmatrix}17&36\\16&7\end{bmatrix}$, $\begin{bmatrix}31&34\\18&23\end{bmatrix}$, $\begin{bmatrix}39&4\\16&39\end{bmatrix}$, $\begin{bmatrix}43&22\\46&21\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	60.240.8-60.e.1.1, 60.240.8-60.e.1.2, 60.240.8-60.e.1.3, 60.240.8-60.e.1.4, 120.240.8-60.e.1.1, 120.240.8-60.e.1.2, 120.240.8-60.e.1.3, 120.240.8-60.e.1.4, 120.240.8-60.e.1.5, 120.240.8-60.e.1.6, 120.240.8-60.e.1.7, 120.240.8-60.e.1.8, 120.240.8-60.e.1.9, 120.240.8-60.e.1.10, 120.240.8-60.e.1.11, 120.240.8-60.e.1.12
Cyclic 60-isogeny field degree:	$48$
Cyclic 60-torsion field degree:	$768$
Full 60-torsion field degree:	$18432$

Jacobian

Conductor:	$2^{18}\cdot3^{8}\cdot5^{16}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{8}$
Newforms:	50.2.a.b$^{2}$, 400.2.a.a, 400.2.a.f, 450.2.a.g$^{2}$, 1800.2.a.r$^{2}$

Models

Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations

$ 0 $	$=$	$ 2 x^{2} - x y + y^{2} + z w + w^{2} $
	$=$	$x^{2} + x y - x z - x t - x u - 2 x r - y z - y t - y u + w t - w u$
	$=$	$x^{2} - x y - x z - x t - x u + y z + y t + y u + y r - w^{2} - w t + w u + w v$
	$=$	$2 x w - x v + y t - y u - z w + z r - w t - w u$
	$=$	$\cdots$

[2*x^2 - x*y + y^2 + z*w + w^2, x^2 + x*y - x*z - x*t - x*u - 2*x*r - y*z - y*t - y*u + w*t - w*u, x^2 - x*y - x*z - x*t - x*u + y*z + y*t + y*u + y*r - w^2 - w*t + w*u + w*v, 2*x*w - x*v + y*t - y*u - z*w + z*r - w*t - w*u, x*w + 2*x*t - 2*x*u - x*v + y*w - y*v + z*r + w*r, x^2 - x*y - x*z - x*t - x*u - x*r + y*z + y*t + y*u + z*w + z*t - z*u - z*v + w*t - w*u, x*z + 2*x*w - x*v - y*w - y*t + y*u + y*v - z^2 - z*w - z*t - z*u - z*r - w*t - w*u - w*r, 2*x^2 + 2*x*y + 2*x*z + x*t + x*u + 2*x*r - 2*y^2 - y*w - 2*y*t + 2*y*u + y*v - y*r + 7*z^2 - 2*z*w - 2*z*r - w^2 - w*v - w*r + t^2 + u^2, 4*x^2 + 2*x*y + x*z - 4*x*w + x*t - 2*x*v + 3*x*r + 2*y^2 - 4*y*z + 4*y*t + 3*y*u + y*r - z^2 - 6*z*w - 3*z*u - z*v + w^2 - 2*w*t - w*u + w*v + t^2 - 3*t*u - t*v - t*r + u*v - u*r, 4*x^2 + 6*x*y - x*z + 2*x*w + x*u + 3*x*v + 2*y^2 + 7*y*z + y*w - y*u - y*v + z^2 - 3*z*w + z*u - z*r + 2*w^2 + 4*w*t + w*u + w*r - 3*t*u - t*v - t*r + u^2 + u*v - u*r, 4*x^2 - 6*x*y + 6*x*z + 2*x*w - x*t + 3*x*v - 2*y^2 - 7*y*z + y*w + y*u - y*v + z^2 - 6*z*w + 2*z*t - z*v - z*r - 2*w^2 + 4*w*t + 2*w*u + w*r + t^2 - t*u - t*v - u*r, 3*x^2 - 4*x*y - 5*x*z - 6*x*w + 3*x*t + 2*x*u - x*v + x*r - y^2 + 5*y*z - 2*y*t - 3*y*u - 2*y*r - 10*z*w - z*t + z*u + z*v + w^2 + w*t - 3*w*u - 2*w*v - t*u - t*r + u^2 + u*v, 10*x^2 + 5*x*y + 2*x*z - 2*x*w + 3*x*t + 3*x*u + x*v + 2*x*r - 5*y^2 + y*w + y*t - y*u - y*v - 2*y*r - 7*z^2 - 3*z*w + z*t - z*u - z*v - 4*w^2 + 2*w*t + w*r - t^2 + t*v - t*r - u^2 - u*v - u*r - v^2 - r^2, 2*x^2 + 2*x*y + 4*x*z + 4*x*w - 3*x*t - 2*x*u + 2*x*v - 12*y^2 - 5*y*z + y*w + 3*y*t + 2*y*u - y*v + 4*y*r - 3*z*w - z*t - z*u - z*r + 15*w^2 + w*t + 3*w*u + 2*w*v + w*r + t^2 - 5*t*u - 3*t*v - t*r + 2*u*v - 2*u*r + v^2 + r^2, x^2 - 5*x*y + 3*x*z + 3*x*t - 4*x*u + 3*x*v - 3*x*r + 5*y*z - 25*y*w + y*t - 6*y*u - 5*y*v - 4*z^2 - 4*z*w + z*t + 5*z*u + z*v + 4*z*r + 6*w*t + 2*w*u + 5*w*r - 2*t^2 - t*u - 2*t*r + u^2 + u*v + u*r]

Singular plane model Singular plane model

$ 0 $

$=$

$ 32896 x^{12} - 252156 x^{11} y - 2048 x^{11} z + 1004768 x^{10} y^{2} + 28184 x^{10} y z + \cdots + 4 y^{4} z^{8} $

[32896*x^12 - 252156*x^11*y - 2048*x^11*z + 1004768*x^10*y^2 + 28184*x^10*y*z + 2304*x^10*z^2 - 2546203*x^9*y^3 - 122912*x^9*y^2*z - 23508*x^9*y*z^2 - 512*x^9*z^3 + 4494495*x^8*y^4 + 292274*x^8*y^3*z + 91982*x^8*y^2*z^2 + 5344*x^8*y*z^3 + 256*x^8*z^4 - 5626186*x^7*y^5 - 488836*x^7*y^4*z - 193962*x^7*y^3*z^2 - 19876*x^7*y^2*z^3 - 2512*x^7*y*z^4 + 4930410*x^6*y^6 + 646480*x^6*y^5*z + 259744*x^6*y^4*z^2 + 32878*x^6*y^3*z^3 + 10222*x^6*y^2*z^4 + 288*x^6*y*z^5 - 2845999*x^5*y^7 - 682172*x^5*y^6*z - 240434*x^5*y^5*z^2 - 28142*x^5*y^4*z^3 - 19947*x^5*y^3*z^4 - 1504*x^5*y^2*z^5 - 56*x^5*y*z^6 + 977611*x^4*y^8 + 521086*x^4*y^7*z + 156362*x^4*y^6*z^2 + 11234*x^4*y^5*z^3 + 22315*x^4*y^4*z^4 + 3008*x^4*y^3*z^5 + 340*x^4*y^2*z^6 - 164640*x^3*y^9 - 244384*x^3*y^8*z - 68432*x^3*y^7*z^2 + 458*x^3*y^6*z^3 - 15877*x^3*y^5*z^4 - 3304*x^3*y^4*z^5 - 680*x^3*y^3*z^6 - 16*x^3*y^2*z^7 + 27900*x^2*y^10 + 50280*x^2*y^9*z + 18248*x^2*y^8*z^2 - 1648*x^2*y^7*z^3 + 6771*x^2*y^6*z^4 + 2448*x^2*y^5*z^5 + 620*x^2*y^4*z^6 + 32*x^2*y^3*z^7 + 4*x^2*y^2*z^8 - 248*x*y^8*z^3 - 1096*x*y^7*z^4 - 1176*x*y^6*z^5 - 336*x*y^5*z^6 - 16*x*y^4*z^7 - 8*x*y^3*z^8 + 124*y^8*z^4 + 240*y^7*z^5 + 112*y^6*z^6 + 4*y^4*z^8]

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=7$, and therefore no rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle t$

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 20.60.4.a.1 :

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

Equation of the image curve:

$0$	$=$	$ X^{2}+XY+2Y^{2}+ZW+W^{2} $
	$=$	$ X^{3}-X^{2}Y-XZ^{2}-XZW-YZW+XW^{2} $

Modular covers

Cover information

Click on a modular curve in the diagram to see information about it.

See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.60.4.a.1	$20$	$2$	$2$	$4$	$1$	$1^{4}$
60.24.0.b.1	$60$	$5$	$5$	$0$	$0$	full Jacobian
60.60.4.d.1	$60$	$2$	$2$	$4$	$0$	$1^{4}$
60.60.4.cj.1	$60$	$2$	$2$	$4$	$1$	$1^{4}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
60.360.22.e.1	$60$	$3$	$3$	$22$	$6$	$1^{14}$
60.360.28.j.1	$60$	$3$	$3$	$28$	$10$	$1^{20}$
60.480.29.da.1	$60$	$4$	$4$	$29$	$7$	$1^{21}$
60.480.35.j.1	$60$	$4$	$4$	$35$	$9$	$1^{27}$
120.240.17.r.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.s.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.ef.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.eg.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.gz.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.ha.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.ht.1	$120$	$2$	$2$	$17$	$?$	not computed
120.240.17.hu.1	$120$	$2$	$2$	$17$	$?$	not computed

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