LMFDB - Modular curve 80.480.13-40.nd.1.1

Invariants

Level:	$80$	$\SL_2$-level:	$80$	Newform level:	$400$
Index:	$480$	$\PSL_2$-index:	$240$
Genus:	$13 = 1 + \frac{ 240 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)	Cusp widths	$5^{8}\cdot10^{4}\cdot40^{4}$	Cusp orbits	$4^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$5 \le \gamma \le 24$
$\overline{\Q}$-gonality:	$5 \le \gamma \le 13$
Rational cusps:	$0$
Rational CM points:	none

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}1&16\\44&37\end{bmatrix}$, $\begin{bmatrix}1&62\\38&33\end{bmatrix}$, $\begin{bmatrix}2&7\\53&4\end{bmatrix}$, $\begin{bmatrix}20&49\\51&34\end{bmatrix}$, $\begin{bmatrix}21&22\\78&53\end{bmatrix}$, $\begin{bmatrix}68&55\\23&12\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.240.13.nd.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree:	$12$
Cyclic 80-torsion field degree:	$192$
Full 80-torsion field degree:	$24576$

Models

Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations

$ 0 $	$=$	$ x b + x d - y r + w r + v r $
	$=$	$x a - t a + t b + t d + u c$
	$=$	$x a + y r + z s - z b$
	$=$	$x a + z s - z d + t d + u r$
	$=$	$\cdots$

[x*b + x*d - y*r + w*r + v*r, x*a - t*a + t*b + t*d + u*c, x*a + y*r + z*s - z*b, x*a + z*s - z*d + t*d + u*r, x*s - x*a - x*b - y*r - w*r + u*r - u*c, x*r + y*b - w*b + w*d - t*r, y*r + y*c - z*s - z*d + t*a - v*r, x*r + y*a - z*r + u*s - u*b - v*s - v*a + v*b, x*s + x*d + y*r - y*c + z*s - z*d + w*r + t*d + u*c + v*c, x*r - y*s - y*b + y*d + w*s - t*c - u*s + u*b, y*s - y*a + y*b + z*r - t*c + u*s - v*b + v*d, x*a - y*r + z*s + z*b - z*d + w*r + t*b - u*r, x*r + z*c + t*r + t*c - u*a, x*s + x*d + z*s + 2*t*s - t*a - u*r + v*r, y*b - y*d - z*r + t*r + u*b - v*b + v*d, x*c + y*s - y*b - z*r - z*c + t*r - u*s + u*b, x*r + y*a - y*b - y*d - z*r - z*c - w*s + w*b + t*r, x*r - 2*x*c + y*b - z*r + z*c - w*s + t*c - u*b + v*b - v*d, y*s + y*b + y*d - z*r - w*b + w*d + t*r + u*s - u*a - u*b - v*a, x*r + x*c + y*s - y*d + z*r + z*c - w*a + w*b + u*s - u*b, x*s - x*b - y*c + z*a - t*a - u*r + u*c, x*r - y*s + z*r - t*r - t*c + 2*u*s - u*a + u*b + u*d + v*s, x*s - x*a + 2*x*b - 2*x*d - u*c, y*c + z*s - z*a + z*b - w*r - w*c - t*b - u*r + u*c + v*r, r^2 - r*c + 2*s^2 - s*d - a^2 + a*b + a*d - 2*b^2 + b*d, 3*r*c + 2*s^2 - 2*s*a - s*d - a*b + 2*a*d + 2*b^2 - b*d + c^2 - d^2, x^2 - 2*x*z + 2*y^2 + 2*y*w - y*v + 2*z^2 - w*v - t^2 + 2*u*v + v^2 - s^2 + s*b - s*d - b^2 + b*d, 2*x^2 + x*z - y^2 - y*u + 2*y*v + z^2 + w*v + 2*u*v + v^2 - r^2 + r*c + s^2 - s*a - s*b + a*b - b^2, x*z - 3*x*t + y^2 - 3*y*u + 2*y*v + z^2 + w*v + t^2 + 2*u^2 - u*v - v^2 + s^2 - s*a - b^2 + b*d, 2*x^2 + 3*x*t + 2*y^2 + y*u + 2*y*v - t^2 - u^2 - u*v + v^2 - r^2 + r*c - s^2 - s*d, x^2 + 4*x*t - 2*y^2 + 2*y*v + 2*t^2 - r^2, x^2 - 2*x*z - y^2 + y*w - 2*y*u - z^2 - w*u - w*v + t^2 - u^2 + 4*u*v + v^2 - r^2, 2*x^2 - x*z + x*t + y^2 - y*w + z^2 - w*u - w*v - 2*u^2 - u*v - r^2 + 2*s^2 - s*b - s*d - a^2 + a*b + a*d, x^2 - x*z + 2*x*t + y^2 + 3*y*w - y*u + z^2 + w*u - 2*t^2 - 2*u^2 - u*v + r^2 - s^2 - s*b + s*d + a*b - a*d - b^2 + d^2, x*z + 2*x*t + 3*y^2 + 2*y*w - 3*y*u - z^2 - w*u - t^2 + u^2 + v^2 + r*c - s^2 - s*b, 2*y^2 + 2*y*w - w^2 - 2*w*u - w*v + u^2 + u*v - v^2 + s^2 - s*a + b*d, 2*x^2 + x*z + y^2 + y*w + z^2 + w^2 + w*u - w*v + u^2 - 2*u*v + r^2 - r*c - 2*s^2 + s*a + 2*s*b - s*d - a*b - b^2 + b*d, 2*x^2 + 3*x*t + y^2 + y*w - y*u + y*v - w^2 - t^2 + u^2 + u*v - v^2 + r^2 - r*c + s^2 - s*a + s*b - b^2 + b*d, x^2 - x*z + 2*x*t - y^2 - y*w + y*u + y*v + z^2 - w^2 + w*u - 2*t^2 + u^2 + 2*u*v + 2*v^2 - r^2 + 2*r*c - s*d + a*d + b^2 - b*d - d^2, 2*x*z - x*t + 2*y^2 - y*w - w^2 + 2*w*u + 3*u^2 - u*v + r*c + s*b - s*d - a*b + a*d + b^2 - d^2, x^2 - x*z + 2*x*t + y^2 + y*w + y*u - y*v - z*t - w^2 - w*v - 2*u*v - v^2 + r*c - s*b + s*d + 2*b^2 - b*d, 2*x^2 - 2*x*t + y^2 - y*v - z^2 + z*t, y^2 + 2*y*w - y*u - 2*y*v + z^2 - w^2 - w*u + w*v + t^2 + u^2 + v^2 + r^2 - 2*r*c - s^2 + s*a + s*b - s*d - 2*b^2 + b*d - c^2, 3*x*y - x*w + y*z - y*t + z*w - z*u + w*t + t*v, 2*x*y - x*w - 3*x*u + x*v + y*t + z*w + 2*z*u - t*u + r*s, x*y + x*w + 2*x*u + x*v - 3*y*t - z*u - t*u - 2*t*v + 2*r*b - r*d - b*c + c*d, 3*x*y - x*u - 3*x*v + y*z - z*w + z*u + w*t - 2*t*u - 2*t*v - r*s + r*a - r*d, 3*x*y - 2*x*w + x*u - 2*x*v + y*t + 2*z*u + 2*t*u - t*v - r*s + r*a - r*b, 4*y*z - 2*y*t + 2*z*w - w*t + r*s - s*c, x*y + x*w + 2*x*u + x*v + 2*y*z + y*t + z*w - z*u + 2*w*t - t*u - 2*t*v + r*b, x*y + x*w + 2*x*u + x*v + y*z - y*t + z*w + z*u + z*v + 2*w*t + 3*t*u + r*d, x*y + 3*x*u - x*v - y*z - y*t + z*u + 2*z*v + w*t - t*u, x*y + 3*x*w + 3*y*z - 2*z*w + z*v - w*t - t*u - t*v + r*s - r*a + r*d - c*d, x*y - 2*x*u - x*v - 3*y*z + z*w - z*u + z*v - 2*w*t - 2*t*v + r*s + r*b - r*d - b*c + c*d, 3*x*y + 4*x*w - y*z + 2*z*w + 2*z*u - z*v - 2*w*t + t*u - t*v + r*b - s*c + a*c - b*c]

Singular plane model Singular plane model

$ 0 $

$=$

$ 2401 x^{8} y^{16} - 26950 x^{8} y^{14} z^{2} + 42795 x^{8} y^{12} z^{4} - 28900 x^{8} y^{10} z^{6} + \cdots + z^{24} $

Rational points

This modular curve has no $\Q_p$ points for $p=3,11,43,67,83$, and therefore no rational points.

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 20.60.3.c.1 :

$\displaystyle X$	$=$	$\displaystyle 5x+2z-t$
$\displaystyle Y$	$=$	$\displaystyle 3z+t$
$\displaystyle Z$	$=$	$\displaystyle -z+3t$

Equation of the image curve:

$0$

$=$

$ 2X^{4}-4X^{3}Y+6X^{2}Y^{2}-4XY^{3}+2Y^{4}+4X^{3}Z+17X^{2}YZ-17XY^{2}Z-4Y^{3}Z+5X^{2}Z^{2}+18XYZ^{2}+5Y^{2}Z^{2}+3XZ^{3}-3YZ^{3}-2Z^{4} $

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.240.13.nd.1 :

$\displaystyle X$	$=$	$\displaystyle d$
$\displaystyle Y$	$=$	$\displaystyle 5x$
$\displaystyle Z$	$=$	$\displaystyle c$

Equation of the image curve:

$0$

$=$

$ 2401X^{8}Y^{16}-5488X^{6}Y^{18}+4018X^{4}Y^{20}-1008X^{2}Y^{22}+81Y^{24}-26950X^{8}Y^{14}Z^{2}+43148X^{6}Y^{16}Z^{2}-25873X^{4}Y^{18}Z^{2}+6828X^{2}Y^{20}Z^{2}-756Y^{22}Z^{2}+42795X^{8}Y^{12}Z^{4}-82152X^{6}Y^{14}Z^{4}+62372X^{4}Y^{16}Z^{4}-22036X^{2}Y^{18}Z^{4}+3186Y^{20}Z^{4}-28900X^{8}Y^{10}Z^{6}+81044X^{6}Y^{12}Z^{6}-91752X^{4}Y^{14}Z^{6}+44928X^{2}Y^{16}Z^{6}-8004Y^{18}Z^{6}+9975X^{8}Y^{8}Z^{8}-47200X^{6}Y^{10}Z^{8}+94216X^{4}Y^{12}Z^{8}-63552X^{2}Y^{14}Z^{8}+13327Y^{16}Z^{8}-1750X^{8}Y^{6}Z^{10}+16820X^{6}Y^{8}Z^{10}-72002X^{4}Y^{10}Z^{10}+63992X^{2}Y^{12}Z^{10}-15464Y^{14}Z^{10}+125X^{8}Y^{4}Z^{12}-3400X^{6}Y^{6}Z^{12}+40252X^{4}Y^{8}Z^{12}-45544X^{2}Y^{10}Z^{12}+12796Y^{12}Z^{12}+300X^{6}Y^{4}Z^{14}-15360X^{4}Y^{6}Z^{14}+22240X^{2}Y^{8}Z^{14}-7592Y^{10}Z^{14}+3430X^{4}Y^{4}Z^{16}-7024X^{2}Y^{6}Z^{16}+3199Y^{8}Z^{16}-325X^{4}Y^{2}Z^{18}+1276X^{2}Y^{4}Z^{18}-932Y^{6}Z^{18}-100X^{2}Y^{2}Z^{20}+178Y^{4}Z^{20}-20Y^{2}Z^{22}+Z^{24} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.240.7-40.cj.1.1	$80$	$2$	$2$	$7$	$?$
80.240.7-40.cj.1.3	$80$	$2$	$2$	$7$	$?$

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