LMFDB - Modular curve 80.480.13-40.nf.1.1

Invariants

Level:	$80$	$\SL_2$-level:	$80$	Newform level:	$1600$
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:	$\begin{bmatrix}4&35\\39&16\end{bmatrix}$, $\begin{bmatrix}14&57\\51&76\end{bmatrix}$, $\begin{bmatrix}42&27\\59&18\end{bmatrix}$, $\begin{bmatrix}46&21\\9&42\end{bmatrix}$, $\begin{bmatrix}47&32\\44&43\end{bmatrix}$, $\begin{bmatrix}71&22\\28&73\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.240.13.nf.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 c + x d - y r + w r - v r $
	$=$	$x a + t a + t c + t d + u b$
	$=$	$x a - y r - z s + z c$
	$=$	$x a - z s + z d + t d - u r$
	$=$	$\cdots$

[x*c + x*d - y*r + w*r - v*r, x*a + t*a + t*c + t*d + u*b, x*a - y*r - z*s + z*c, x*a - z*s + z*d + t*d - u*r, x*s + x*a - x*c - y*r - w*r + u*r + u*b, x*r + y*c - w*c + w*d + t*r, y*r - y*b - z*s - z*d + t*a + v*r, x*a + y*r - z*s - z*c + z*d - w*r + t*c + u*r, x*r - y*a - z*r + u*s - u*c + v*s - v*a - v*c, x*s + x*d + y*r + y*b + z*s - z*d + w*r - t*d - u*b + v*b, x*r - y*s - y*c + y*d + w*s - t*b - u*s + u*c, y*s + y*a + y*c + z*r - t*b + u*s + v*c - v*d, x*s + x*d + z*s - 2*t*s - t*a - u*r - v*r, x*r - z*b - t*r + t*b + u*a, y*c - y*d - z*r - t*r + u*c + v*c - v*d, x*b - y*s + y*c + z*r - z*b + t*r + u*s - u*c, x*r - y*a - y*c - y*d - z*r + z*b - w*s + w*c - t*r, x*r + 2*x*b + y*c - z*r - z*b - w*s + t*b - u*c - v*c + v*d, y*s + y*c + y*d - z*r - w*c + w*d - t*r + u*s + u*a - u*c - v*a, x*r - x*b + y*s - y*d + z*r - z*b + w*a + w*c + u*s - u*c, x*s - x*c + y*b - z*a - t*a - u*r - u*b, x*r - y*s + z*r + t*r - t*b + 2*u*s + u*a + u*c + u*d - v*s, x*s + x*a + 2*x*c - 2*x*d + u*b, y*b - z*s - z*a - z*c + w*r - w*b - t*c + u*r + u*b + v*r, r^2 + r*b + 2*s^2 - s*d - a^2 - a*c - a*d - 2*c^2 + c*d, 2*x^2 + 2*x*t + y^2 + y*v - z^2 - z*t, x^2 + 3*x*z + y^2 + y*u + y*v - z^2 + w*u + t^2, 2*x^2 + x*z - y^2 + y*w - 2*y*u - y*v + 2*z^2 - z*t + w*v - 2*t^2, 2*x^2 - 3*x*t - 2*y^2 + y*w + y*u - 2*y*v - 2*w*u + w*v - t^2, 3*x*y - x*w + y*z + y*t + z*w - z*u - w*t + t*v, x*y + 3*x*u + x*v - y*z + y*t + z*u - 2*z*v - w*t + t*u, x^2 + x*t - y^2 + y*u + 2*z^2 + 2*z*t + w^2 - 2*w*u + w*v, 3*r*b - 2*s^2 - 2*s*a + s*d - a*c + 2*a*d - b^2 - 2*c^2 + c*d + d^2, x*y - 3*x*w + x*u - 3*x*v + y*z + 2*z*u + 2*z*v + w*t + t*u + t*v, x^2 - x*z - 2*x*t - y^2 - y*w + y*u - 2*y*v + z^2 - z*t - w*v + u^2 + 4*u*v - v^2, x^2 - x*z - 2*x*t + 2*y^2 + 3*y*w - 2*z^2 + w^2 + w*u + 2*w*v + u^2 + v^2 - r^2 - r*b + 2*s^2 + s*a - 2*s*c + s*d - a*c + c^2 - c*d, x^2 - x*z - 2*x*t + 2*y^2 + 3*y*w - 2*z^2 + w^2 + w*u + 2*w*v + u^2 + v^2 + r^2 + r*b + s*a + 4*s*c - 2*s*d + a^2 + 3*a*c - a*d + c^2 - d^2, 2*x^2 - x*z - x*t + 3*y^2 + y*u + y*v + 4*z^2 - 3*z*t - 2*w^2 + 2*w*u - 2*w*v - t^2 + r^2 + r*b - s^2 - s*a + s*c + a*c, x^2 - 2*x*z - y^2 + 4*y*w + 3*y*v + 2*z^2 - 3*z*t - w^2 + w*u + w*v + 3*u^2 - 2*u*v - v^2 - r^2 - r*b - s*c + c^2 - c*d, 2*x^2 + 3*x*z + x*t - y*w + 3*y*u - 3*y*v + z^2 + z*t - w*u - 3*w*v - 2*t^2 + u^2 + v^2 + r*b - s^2 - s*a + s*c + a*c + c^2, x^2 + x*t - y^2 + y*u - w^2 + 2*w*u - w*v + 2*t^2 - 2*u^2 + 2*u*v + 2*v^2 + r^2 + 2*r*b - 2*s^2 + 3*s*c + a^2 + 2*a*c + a*d + c*d, x^2 + x*t - 2*y*w - y*u + y*v + z^2 + z*t + w^2 + 2*w*u - w*v + 6*t^2 + r^2, 4*x^2 - x*t + 2*y^2 + y*w - y*u + 2*z^2 + 2*z*t - 2*w^2 + 2*w*u - w*v + t^2 + r^2 + 2*r*b - s^2 - s*a + 2*s*c - s*d + a*c - c^2 + c*d, 2*x*w - 2*x*u + x*v + 3*y*t - z*w + z*u - z*v - w*t + t*v, x^2 + x*z - x*t + 2*y*w + y*u - 2*y*v + 2*z^2 + 3*w^2 + w*u - 3*w*v - t^2 - 3*u^2 + 2*u*v + v^2 - r^2 + 2*r*b - s^2 - 2*s*a + s*d + a*d - b^2 + c^2, x^2 - x*z - 2*x*t - 2*y^2 - y*w + 2*z*t + 5*w^2 + w*u + 2*w*v - 2*t^2 + u^2 + v^2 - r^2 + r*b - s^2 + s*c + a^2 + a*c + a*d - b^2 + c^2, x^2 - x*z - 2*x*t - y^2 + 4*y*v - z^2 + w^2 - 2*w*u - 3*w*v + t^2 + 2*u^2 + 2*v^2 + r*b + 2*s*c - s*d + a*c + c*d, 2*x*y - x*w - 3*x*u - x*v - 2*y*z - 2*y*t + z*w + 6*z*u - 2*z*v + 3*t*u - t*v - r*s, x*y + 2*x*w + x*u - 3*x*v + y*z + 5*y*t - 3*z*u + 2*z*v + w*t + t*u - 4*t*v - 2*r*c + r*d - b*c + b*d, 2*x*w - 2*x*u + x*v + 4*y*z - 5*y*t + z*w + z*u - z*v - 5*w*t + t*v + r*c - r*d + b*c - b*d, 2*x*y - x*w - 3*x*u + 4*x*v - y*t - 2*z*w + 5*z*u + z*v + w*t - 5*t*u - 2*t*v + r*s + r*a + r*c, 2*x*y + x*u - 3*x*v - 4*y*z + y*t + 3*z*w + 3*z*u - 2*z*v + 2*w*t - 5*t*u - 4*t*v - r*s - r*c - b*c + b*d, 3*x*y + x*w - 2*x*u + x*v + 3*y*z - 4*z*w - z*v + 6*w*t + 2*t*v - r*s - r*a + r*c - r*d - b*d, 3*x*y - x*w - 7*y*z - 3*y*t - 3*z*w - z*u - 3*w*t + t*v + r*s + s*b, 4*x*y + 8*x*w - x*u + 3*x*v + 2*y*z - y*t + z*w + z*u + 4*z*v + 2*w*t + 3*t*u + 2*t*v + r*s - s*b - a*b - b*d]

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3,11,13,43,67,83$, 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} $

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.1.15	$80$	$2$	$2$	$7$	$?$

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