LMFDB - Modular curve 80.480.13-40.of.1.2

Invariants

Level:	$80$	$\SL_2$-level:	$80$	Newform level:	$200$
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:	$\begin{bmatrix}4&43\\77&26\end{bmatrix}$, $\begin{bmatrix}26&31\\7&34\end{bmatrix}$, $\begin{bmatrix}33&74\\42&57\end{bmatrix}$, $\begin{bmatrix}41&32\\58&79\end{bmatrix}$, $\begin{bmatrix}46&53\\29&14\end{bmatrix}$, $\begin{bmatrix}69&14\\28&31\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.240.13.of.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 t + x v - y t - y u + a b + a c $
	$=$	$2 x^{2} + x z + x w + x a + y^{2} + y w - z a + v s$
	$=$	$x a + 3 y^{2} - y z - y w - z a - r s + c^{2}$
	$=$	$x^{2} + 3 x y + y^{2} + v s + r s + b c$
	$=$	$\cdots$

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

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3,13,17,19,43,67,83$, 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 :

$\displaystyle X$	$=$	$\displaystyle -5x-5y+2z+2w+a$
$\displaystyle Y$	$=$	$\displaystyle 3z+3w-a$
$\displaystyle Z$	$=$	$\displaystyle -z-w-3a$

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} $

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank
5.20.0.b.1	$5$	$24$	$12$	$0$	$0$
16.24.0-8.n.1.8	$16$	$20$	$20$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.120.4-40.bl.1.5	$80$	$4$	$4$	$4$	$?$
80.240.7-40.cj.1.1	$80$	$2$	$2$	$7$	$?$
80.240.7-40.cj.1.4	$80$	$2$	$2$	$7$	$?$

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