LMFDB - Modular curve 80.480.13-40.ol.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:	$\begin{bmatrix}8&57\\75&22\end{bmatrix}$, $\begin{bmatrix}13&66\\64&39\end{bmatrix}$, $\begin{bmatrix}30&67\\27&70\end{bmatrix}$, $\begin{bmatrix}33&64\\36&77\end{bmatrix}$, $\begin{bmatrix}53&30\\40&3\end{bmatrix}$, $\begin{bmatrix}74&65\\55&44\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.240.13.ol.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 a - x b - x c - y s + r s $
	$=$	$x a + x c - x d - v s$
	$=$	$x c - x d + t b - t c$
	$=$	$x b + x c - y u + z c$
	$=$	$\cdots$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 24010000 x^{8} y^{16} + 107800000 x^{8} y^{14} z^{2} + 68472000 x^{8} y^{12} z^{4} + 18496000 x^{8} y^{10} z^{6} + \cdots + 4096 z^{24} $

Rational points

This modular curve has no real points and no $\Q_p$ points for $p=3,11,17$, 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.ol.1 :

$\displaystyle X$	$=$	$\displaystyle d$
$\displaystyle Y$	$=$	$\displaystyle 2x$
$\displaystyle Z$	$=$	$\displaystyle s$

Equation of the image curve:

$0$

$=$

$ 24010000X^{8}Y^{16}+274400000X^{6}Y^{18}+790125000X^{4}Y^{20}+35000000X^{2}Y^{22}+390625Y^{24}+107800000X^{8}Y^{14}Z^{2}+216160000X^{6}Y^{16}Z^{2}-503375000X^{4}Y^{18}Z^{2}-864000000X^{2}Y^{20}Z^{2}+5625000Y^{22}Z^{2}+68472000X^{8}Y^{12}Z^{4}+40960000X^{6}Y^{14}Z^{4}-350480000X^{4}Y^{16}Z^{4}+1070400000X^{2}Y^{18}Z^{4}+29125000Y^{20}Z^{4}+18496000X^{8}Y^{10}Z^{6}+21248000X^{6}Y^{12}Z^{6}+133520000X^{4}Y^{14}Z^{6}+336640000X^{2}Y^{16}Z^{6}+58900000Y^{18}Z^{6}+2553600X^{8}Y^{8}Z^{8}+4608000X^{6}Y^{10}Z^{8}+307296000X^{4}Y^{12}Z^{8}-243200000X^{2}Y^{14}Z^{8}+13110000Y^{16}Z^{8}+179200X^{8}Y^{6}Z^{10}+307200X^{6}Y^{8}Z^{10}+146656000X^{4}Y^{10}Z^{10}-83968000X^{2}Y^{12}Z^{10}-65760000Y^{14}Z^{10}+5120X^{8}Y^{4}Z^{12}+33126400X^{4}Y^{8}Z^{12}+15360000X^{2}Y^{10}Z^{12}+4192000Y^{12}Z^{12}+4147200X^{4}Y^{6}Z^{14}+5734400X^{2}Y^{8}Z^{14}+13440000Y^{10}Z^{14}+296960X^{4}Y^{4}Z^{16}+409600X^{2}Y^{6}Z^{16}-569600Y^{8}Z^{16}+10240X^{4}Y^{2}Z^{18}-1177600Y^{6}Z^{18}-30720Y^{4}Z^{20}+40960Y^{2}Z^{22}+4096Z^{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.1.16	$80$	$2$	$2$	$7$	$?$

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