LMFDB - Modular curve 16.384.21.bc.1

Invariants

Level:	$16$	$\SL_2$-level:	$16$	Newform level:	$256$
Index:	$384$	$\PSL_2$-index:	$384$
Genus:	$21 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$
Cusps:	$24$ (of which $8$ are rational)	Cusp widths	$16^{24}$	Cusp orbits	$1^{8}\cdot4^{2}\cdot8$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$8$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	16E21
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	16.384.21.9

Level structure

$\GL_2(\Z/16\Z)$-generators:	$\begin{bmatrix}1&8\\4&15\end{bmatrix}$, $\begin{bmatrix}11&8\\0&15\end{bmatrix}$, $\begin{bmatrix}13&0\\4&15\end{bmatrix}$, $\begin{bmatrix}15&4\\4&15\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup:	$C_4^2:C_2^2$
Contains $-I$:	yes
Quadratic refinements:	16.768.21-16.bc.1.1, 16.768.21-16.bc.1.2, 16.768.21-16.bc.1.3, 16.768.21-16.bc.1.4, 16.768.21-16.bc.1.5, 16.768.21-16.bc.1.6, 32.768.21-16.bc.1.1, 32.768.21-16.bc.1.2, 32.768.21-16.bc.1.3, 32.768.21-16.bc.1.4, 32.768.21-16.bc.1.5, 32.768.21-16.bc.1.6, 32.768.21-16.bc.1.7, 32.768.21-16.bc.1.8, 32.768.21-16.bc.1.9, 32.768.21-16.bc.1.10, 48.768.21-16.bc.1.1, 48.768.21-16.bc.1.2, 48.768.21-16.bc.1.3, 48.768.21-16.bc.1.4, 48.768.21-16.bc.1.5, 48.768.21-16.bc.1.6
Cyclic 16-isogeny field degree:	$4$
Cyclic 16-torsion field degree:	$8$
Full 16-torsion field degree:	$64$

Jacobian

Conductor:	$2^{146}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{3}\cdot2^{5}\cdot8$
Newforms:	16.2.e.a$^{2}$, 64.2.a.a, 64.2.b.a, 256.2.a.b, 256.2.a.c, 256.2.a.e, 256.2.b.b, 256.2.e.a

Models

Canonical model in $\mathbb{P}^{ 20 }$ defined by 171 equations

$ 0 $	$=$	$ r a + k m $
	$=$	$y r - g m$
	$=$	$y b + g l$
	$=$	$r a + i l + i m + k l$
	$=$	$\cdots$

[r*a + k*m, y*r - g*m, y*b + g*l, r*a + i*l + i*m + k*l, x*i - y*e - h*k, x*k + y*c - h*i, x*i + y*c - g*k, x*k + y*e - g*i, d*k + f*i, x*l + x*m - y*b - h*m, x*l - x*m + y*r + h*l, y*t - g^2 + g*h, z*t - g^2 - g*h, t*g + t*h - f*i, x*c + x*f - f*h, x*e - x*f - f*g, r*a - e^2 + e*f, x*d + x*e + d*h, x*c + x*d - d*g, c*m - e*l, r*l + b*m, r*c + b*e, x*d - x*f - y*m + e*h, r*c - b*e + l^2 - m^2, y*t + r*e + g^2 - g*h - i*k - k^2 - m^2, y*t - z*t + r*c + 2*h^2 - m^2, 2*x^2 - g^2 - h^2, t*g - t*h - r*m - e*k, y*l + d*g + e*g - f*g, t*g - t*h + r*l - e*i + f*i + f*k, 2*x*t + r*l - e*i + f*i, d*l + d*m - f*l + f*m, t*g - t*h - r*l + d*i + e*i - f*i, c*k + d*k + e*i - f*i, c*i - d*k - e*k + f*i, x*c - x*e + c*h + e*g, x*c + x*e + c*g - e*h, r*a + c*f + d*e + i*l, c*e + d*e - i*m, r*i - r*k + e*l - e*m - f*l + f*m, y*u - y*v - z*u + w*v + d*f, r*i - r*k + c*l + e*l + f*l - f*m, 2*y*w + d*e - d*f, c*f - d*e + 2*d*f, c*d - c*f + d*e + e*f, c^2 + c*d - i*l, r*m + b*l + c*i + e*k, t*g - t*h + r*l - a*m, a*l - e*k + f*i + f*k, r*c - a*e - g^2 + h^2, a*d - g^2 + h^2, a*b - k*l, r*l - s*m - e*i, t*g - t*h - s*l, r*i - r*k - b*i - b*k, r*c + b*d - b*f - m^2, y*k + a*g, 2*x*h + y*t - z*t + b*f, b*c + b*d + b*f + l*m, x*b + a*h + b*h, a*d - b*c - b*e + g^2 - h^2 - i^2 + i*k, 2*x*g - y*t + z*t - r*c - b*f - i*k, z*t - a*f - b*d - b*f + g*h + h^2 - i*k + l^2, r*c - a*c - a*f + l^2 + l*m, s*f - a*f + b*e - b*f - i*k, s*f - a*c + b*f - k^2, r*k - s*k - a*i, x*r - s*h, r*c - s*e + s*f + b*f + i*k, r*c + s*d + b*d + g^2 - h^2 + i*k, s*c + g^2 - h^2, s*b - d*e + d*f, r*f - a*f + b*e - b*f + g^2 - h^2 - i*k, z*u - w*v + s*a + c*f, 2*x*z + 2*x*w + a*i - e*l, 2*x*y + 2*y*h + a*i - e*l + f*l, 2*x*z - 2*x*v - 2*y*h - s*k - f*l + f*m, 2*x*z + 2*x*u - 2*y*g - s*k, z*u + 2*w^2 + w*v - r^2 + s*a - e*f + k*l, 2*x*z - 2*x*v + 2*y*h - s*k - d*l + f*l, 2*x*w - 2*x*v - 2*y*h + b*k + c*l, 2*y*g + 2*y*h + r*i - r*k + a*i + a*k, 2*x*y + 2*y*g + r*k - s*i - d*m - e*l, x*s - x*a - s*g, x*r + y*i - s*g - a*h, x*r - r*g - a*h, y*t - z*t + r*c - r*d + b*d - 2*g*h - m^2, r*a + r*b + d*e - d*f, x*y + x*z + v*h + a*i - e*l, x*y - x*z - v*g, x*y - x*z + u*h, r^2 - r*s + i*m, v*m + d*h - f*h, v*l - d*h - f*h, x*r - x*b + v*k, x*r + x*b + v*i, y*b - v*a + h*m, u*m - d*g + f*g, u*l + d*g + f*g, x*y + x*z - u*g, v*k + r*h + b*g, x*s + x*a - y*k - r*h, z*u - w*v + r^2 - s^2 - s*a + a^2 + c*f, x*k - y*f + v*e - v*f, 2*y*z + s*a - a^2 + c*f + d*f - k*l, w*c + w*e - v*e - h*i + h*k, x*i + y*d - v*c - v*d, w*c - w*e + v*c - h*i - h*k, y*b - z*b + v*b - h*l + h*m, z*r + v*s - h*m, y*r - v*r + v*s + h*l, y*k + u*k - s*g - b*g, y*k - u*i + s*g - b*g, x*i - y*f - u*f, x*i + x*k - u*e, x*k - y*d + u*d, x*i - x*k + u*c, y*r - z*b - u*b, y*b + z*r - u*r, y*b + t*f + h*m, x*r - t*m - r*h - a*h, t*l + b*g - b*h, t*k - c*g - e*g, t*i + c*g - e*g, y*r + t*e + h*m, y*r - t*d + h*l, y*b - t*c - h*l, t*b - h*i + h*k, y*c + y*e + t*a, x*i + x*k - t*s, y*t - z*t - t*v + b*f, y*t - z*t - t*u + r*c + b*f + i*k, t^2 + d*e - d*f, t*r - h*i - h*k, w*m - e*h, w*l - c*h, x*r + w*k - b*g, x*r + w*i + b*h, x*v + y*h - w*h, x*u - y*g + w*g, w*e - w*f + h*k, w*c + w*d - h*i, w*b + v*b + h*m, x*m - y*b - w*a, x*l + w*s - v*s, y*r - w*r + v*s, w*t + g*h + h^2, w*u + w*v + d*f, z*m + e*g, z*l + c*g, x*u - y*g - z*h, x*r + y*k - z*k - a*h - b*g, z*i - s*g - b*g, x*u + y*h - z*g, x*k - z*f, x*i - z*e + h*i, x*i + z*d, x*k + z*c - h*k, z*v + w*v + d*f, u^2 - v^2 + s*a - a^2 + d*e + e*f - k*l, y^2 + z^2 - u^2, y*z - y*w - z^2 - z*w, y^2 - y*z - y*w + z*w - u*v, 2*w^2 + u*v - v^2 + d*e - d*f, y*r - y*a - y*b + z*a, y*s + y*a + y*b + u*a, y*s + y*b + z*r - z*s, y*a + y*b - z*s + u*s, w*u - w*v - u*v + v^2 - b^2 + c*d - k*l, y*u + y*v - u*v - v^2 + s*a + d*e + f^2, y*u + y*v + u*v + v^2 + s*a - d^2 + d*e]

Rational points

This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:-1/2:0:0:0:1/2:1/2:0:0:0:0:0:0:0:1:0:0:0:0:0:0)$, $(0:-1/2:0:0:0:-1/2:-1/2:0:0:0:0:0:1:0:0:0:0:0:0:0:0)$, $(0:1/2:0:0:0:1/2:1/2:0:0:0:0:0:1:0:0:0:0:0:0:0:0)$, $(0:0:-1/2:1/2:0:1/2:-1/2:0:0:0:1:0:0:0:0:0:0:0:0:0:0)$, $(0:1/2:0:0:0:-1/2:-1/2:0:0:0:0:0:0:0:1:0:0:0:0:0:0)$, $(0:0:1/2:-1/2:0:1/2:-1/2:1:1:0:0:0:0:0:0:0:0:0:0:0:0)$, $(0:0:-1/2:1/2:0:-1/2:1/2:1:1:0:0:0:0:0:0:0:0:0:0:0:0)$, $(0:0:1/2:-1/2:0:-1/2:1/2:0:0:0:1:0:0:0:0:0:0:0:0:0:0)$

Canonical model

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

Maps to other modular curves

Map of degree 8 from the canonical model of this modular curve to the canonical model of the modular curve 16.48.3.b.1 :

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

Equation of the image curve:

$0$

$=$

$ 4X^{4}+Y^{3}Z+YZ^{3} $

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
16.192.7.r.1	$16$	$2$	$2$	$7$	$0$	$1^{2}\cdot2^{2}\cdot8$
16.192.9.bo.1	$16$	$2$	$2$	$9$	$1$	$2^{2}\cdot8$
16.192.11.b.1	$16$	$2$	$2$	$11$	$0$	$1^{2}\cdot2^{4}$

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
16.768.41.n.1	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{4}\cdot8$
16.768.41.o.1	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{4}\cdot8$
32.768.45.s.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.ca.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.cm.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.cu.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.49.ff.1	$32$	$2$	$2$	$49$	$9$	$2^{6}\cdot4^{4}$
32.768.49.fg.1	$32$	$2$	$2$	$49$	$3$	$2^{4}\cdot4^{3}\cdot8$
32.768.53.e.1	$32$	$2$	$2$	$53$	$1$	$16^{2}$
32.768.53.e.2	$32$	$2$	$2$	$53$	$1$	$16^{2}$
32.768.53.f.1	$32$	$2$	$2$	$53$	$1$	$16^{2}$
32.768.53.f.2	$32$	$2$	$2$	$53$	$1$	$16^{2}$
48.768.41.wx.1	$48$	$2$	$2$	$41$	$6$	$1^{4}\cdot2^{4}\cdot8$
48.768.41.xb.1	$48$	$2$	$2$	$41$	$4$	$1^{4}\cdot2^{4}\cdot8$
48.1152.85.sw.1	$48$	$3$	$3$	$85$	$6$	$1^{12}\cdot2^{6}\cdot4^{2}\cdot8^{4}$
48.1536.105.ll.1	$48$	$4$	$4$	$105$	$13$	$1^{14}\cdot2^{13}\cdot4^{3}\cdot8^{4}$

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