LMFDB - Modular curve 16.384.21.t.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 $4$ are rational)	Cusp widths	$16^{24}$	Cusp orbits	$1^{4}\cdot2^{4}\cdot4^{3}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 8$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	16B21
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	16.384.21.1

Level structure

$\GL_2(\Z/16\Z)$-generators:	$\begin{bmatrix}1&10\\0&7\end{bmatrix}$, $\begin{bmatrix}7&0\\8&7\end{bmatrix}$, $\begin{bmatrix}7&12\\0&3\end{bmatrix}$, $\begin{bmatrix}7&14\\0&1\end{bmatrix}$, $\begin{bmatrix}9&0\\0&1\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup:	$C_2^4\times C_4$
Contains $-I$:	yes
Quadratic refinements:	16.768.21-16.t.1.1, 16.768.21-16.t.1.2, 16.768.21-16.t.1.3, 16.768.21-16.t.1.4, 16.768.21-16.t.1.5, 16.768.21-16.t.1.6, 16.768.21-16.t.1.7, 16.768.21-16.t.1.8, 16.768.21-16.t.1.9, 16.768.21-16.t.1.10, 16.768.21-16.t.1.11, 16.768.21-16.t.1.12, 48.768.21-16.t.1.1, 48.768.21-16.t.1.2, 48.768.21-16.t.1.3, 48.768.21-16.t.1.4, 48.768.21-16.t.1.5, 48.768.21-16.t.1.6, 48.768.21-16.t.1.7, 48.768.21-16.t.1.8, 48.768.21-16.t.1.9, 48.768.21-16.t.1.10, 48.768.21-16.t.1.11, 48.768.21-16.t.1.12
Cyclic 16-isogeny field degree:	$2$
Cyclic 16-torsion field degree:	$8$
Full 16-torsion field degree:	$64$

Jacobian

Conductor:	$2^{156}$
Simple:	no
Squarefree:	no
Decomposition:	$1^{3}\cdot2\cdot8^{2}$
Newforms:	32.2.a.a$^{2}$, 64.2.a.a, 64.2.b.a, 256.2.e.a, 256.2.e.b

Models

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

$ 0 $	$=$	$ x v + i l $
	$=$	$h^{2} - l m$
	$=$	$x z + y c - l m - m^{2}$
	$=$	$x z - x w + x u + k l - l m$
	$=$	$\cdots$

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

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

Maps to other modular curves

Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve $X_{\mathrm{sp}}(8)$ :

$\displaystyle X$	$=$	$\displaystyle h+i$
$\displaystyle Y$	$=$	$\displaystyle h-i$
$\displaystyle Z$	$=$	$\displaystyle k$

Equation of the image curve:

$0$

$=$

$ X^{3}Y+XY^{3}+2Z^{4} $

Modular covers

Cover information

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

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
8.192.5.d.1	$8$	$2$	$2$	$5$	$0$	$8^{2}$
16.192.11.a.2	$16$	$2$	$2$	$11$	$0$	$1^{2}\cdot8$
16.192.11.b.1	$16$	$2$	$2$	$11$	$0$	$1^{2}\cdot8$

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.c.1	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{8}$
16.768.41.e.2	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{8}$
16.768.41.i.1	$16$	$2$	$2$	$41$	$1$	$1^{6}\cdot2^{7}$
16.768.41.i.3	$16$	$2$	$2$	$41$	$1$	$1^{6}\cdot2^{7}$
16.768.41.i.5	$16$	$2$	$2$	$41$	$1$	$1^{6}\cdot2^{7}$
16.768.41.i.7	$16$	$2$	$2$	$41$	$1$	$1^{6}\cdot2^{7}$
16.768.41.k.2	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{8}$
16.768.41.o.1	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{8}$
48.768.41.ey.2	$48$	$2$	$2$	$41$	$6$	$1^{4}\cdot2^{8}$
48.768.41.fm.1	$48$	$2$	$2$	$41$	$6$	$1^{4}\cdot2^{8}$
48.768.41.vc.1	$48$	$2$	$2$	$41$	$2$	$1^{6}\cdot2^{7}$
48.768.41.vc.3	$48$	$2$	$2$	$41$	$2$	$1^{6}\cdot2^{7}$
48.768.41.vc.5	$48$	$2$	$2$	$41$	$2$	$1^{6}\cdot2^{7}$
48.768.41.vc.7	$48$	$2$	$2$	$41$	$2$	$1^{6}\cdot2^{7}$
48.768.41.vq.1	$48$	$2$	$2$	$41$	$6$	$1^{4}\cdot2^{8}$
48.768.41.we.1	$48$	$2$	$2$	$41$	$6$	$1^{4}\cdot2^{8}$
48.1152.85.na.1	$48$	$3$	$3$	$85$	$3$	$1^{16}\cdot2^{6}\cdot4\cdot8^{4}$
48.1536.105.hy.1	$48$	$4$	$4$	$105$	$5$	$1^{18}\cdot2^{7}\cdot4^{5}\cdot8^{4}$

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