LMFDB - Modular curve 16.384.21.be.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:	16D21
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	16.384.21.11

Level structure

$\GL_2(\Z/16\Z)$-generators:	$\begin{bmatrix}1&12\\8&15\end{bmatrix}$, $\begin{bmatrix}7&0\\4&1\end{bmatrix}$, $\begin{bmatrix}7&8\\8&15\end{bmatrix}$, $\begin{bmatrix}11&4\\8&1\end{bmatrix}$
$\GL_2(\Z/16\Z)$-subgroup:	$C_4^2:C_2^2$
Contains $-I$:	yes
Quadratic refinements:	16.768.21-16.be.1.1, 16.768.21-16.be.1.2, 16.768.21-16.be.1.3, 16.768.21-16.be.1.4, 16.768.21-16.be.1.5, 16.768.21-16.be.1.6, 32.768.21-16.be.1.1, 32.768.21-16.be.1.2, 32.768.21-16.be.1.3, 32.768.21-16.be.1.4, 32.768.21-16.be.1.5, 32.768.21-16.be.1.6, 48.768.21-16.be.1.1, 48.768.21-16.be.1.2, 48.768.21-16.be.1.3, 48.768.21-16.be.1.4, 48.768.21-16.be.1.5, 48.768.21-16.be.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.a, 256.2.a.d, 256.2.b.a, 256.2.b.c, 256.2.e.a

Models

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

$ 0 $	$=$	$ v d - k l $
	$=$	$v r + i k$
	$=$	$y b + h i$
	$=$	$x v - g k$
	$=$	$\cdots$

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

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

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 16.96.5.j.1 :

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

Equation of the image curve:

$0$	$=$	$ 2YZ-WT $
	$=$	$ YW+ZW+YT-ZT $
	$=$	$ 2X^{2}+YW-ZT $

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.bp.2	$16$	$2$	$2$	$9$	$1$	$2^{2}\cdot8$
16.192.11.b.2	$16$	$2$	$2$	$11$	$0$	$1^{2}\cdot2^{4}$

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.2	$16$	$2$	$2$	$41$	$2$	$1^{4}\cdot2^{4}\cdot8$
32.768.45.b.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.bx.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.cf.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.ch.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.cr.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
32.768.45.cz.1	$32$	$2$	$2$	$45$	$1$	$4^{2}\cdot8^{2}$
48.768.41.wz.1	$48$	$2$	$2$	$41$	$6$	$1^{4}\cdot2^{4}\cdot8$
48.768.41.xd.1	$48$	$2$	$2$	$41$	$4$	$1^{4}\cdot2^{4}\cdot8$
48.1152.85.tk.1	$48$	$3$	$3$	$85$	$8$	$1^{8}\cdot2^{10}\cdot4\cdot8^{4}$
48.1536.105.lv.1	$48$	$4$	$4$	$105$	$9$	$1^{14}\cdot2^{13}\cdot4^{3}\cdot8^{4}$

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