Modular curve 80.288.7-40.ee.1.16

• Label: 80.288.7-40.ee.1.16
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$1600$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $4$ are rational)		Cusp widths	$2^{4}\cdot8^{2}\cdot10^{4}\cdot40^{2}$		Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 7$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 7$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40M7

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}9&0\\15&3\end{bmatrix}$, $\begin{bmatrix}21&0\\54&73\end{bmatrix}$, $\begin{bmatrix}27&20\\27&71\end{bmatrix}$, $\begin{bmatrix}53&40\\12&1\end{bmatrix}$, $\begin{bmatrix}61&20\\26&79\end{bmatrix}$, $\begin{bmatrix}71&0\\64&3\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.ee.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree:	$2$
Cyclic 80-torsion field degree:	$32$
Full 80-torsion field degree:	$40960$

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $	$=$	$ x^{2} - y t $
	$=$	$x v + 2 w t - t u$
	$=$	$2 x w + x u - z t$
	$=$	$2 x w - x u + y v$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ - 5 x^{8} z^{2} - 200 x^{6} y^{4} - 20 x^{6} y^{2} z^{2} + 50 x^{4} y^{4} z^{2} + 2 x^{4} y^{2} z^{4} + \cdots + 2 y^{6} z^{4} $

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model

$(0:0:-2:1/2:0:1:0)$, $(0:0:0:-1/4:0:1/2:1)$, $(0:0:0:1/4:0:-1/2:1)$, $(0:0:2:1/2:0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -2^6\,\frac{15625z^{12}-187500z^{11}v+656250z^{10}v^{2}-1187500z^{9}v^{3}+2109375z^{8}v^{4}-4125000z^{7}v^{5}+30183413z^{6}v^{6}-123220900z^{5}v^{7}+145053880z^{4}v^{8}-33904484z^{3}v^{9}+293260029z^{2}v^{10}+4086zu^{10}v+1083258zu^{8}v^{3}+160392412zu^{6}v^{5}-14547614zu^{4}v^{7}-651275562zu^{2}v^{9}+86192604zv^{11}-64000000wu^{11}+320025284wu^{9}v^{2}-422307088wu^{7}v^{4}-873448316wu^{5}v^{6}+2225742704wu^{3}v^{8}-815911144wuv^{10}+160039060t^{2}u^{10}-640043230t^{2}u^{8}v^{2}+415672290t^{2}u^{6}v^{4}+2925351460t^{2}u^{4}v^{6}-3046279790t^{2}u^{2}v^{8}+884619850t^{2}v^{10}-31999999u^{12}+144016548u^{10}v^{2}-114057644u^{8}v^{4}-495278393u^{6}v^{6}+1053151493u^{4}v^{8}-744668535u^{2}v^{10}+88461986v^{12}}{v(16z^{6}v^{5}+556z^{5}v^{6}-3147z^{4}v^{7}+10560z^{3}v^{8}-8936z^{2}v^{9}-16zu^{10}-748zu^{8}v^{2}+1709zu^{6}v^{4}-5202zu^{4}v^{6}-7739zu^{2}v^{8}+9960zv^{10}+384wu^{9}v+3660wu^{7}v^{3}-10276wu^{5}v^{5}+32264wu^{3}v^{7}-37352wuv^{9}-1000t^{2}u^{8}v-10410t^{2}u^{6}v^{3}+11500t^{2}u^{4}v^{5}-60770t^{2}u^{2}v^{7}-29450t^{2}v^{9}+192u^{10}v+1274u^{8}v^{3}-1819u^{6}v^{5}+4781u^{4}v^{7}-4257u^{2}v^{9}-2945v^{11})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.ee.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle y$
$\displaystyle Z$	$=$	$\displaystyle 2w$

Equation of the image curve:

$0$

$=$

$ -5X^{8}Z^{2}-200X^{6}Y^{4}-20X^{6}Y^{2}Z^{2}+50X^{4}Y^{4}Z^{2}+2X^{4}Y^{2}Z^{4}-20X^{2}Y^{6}Z^{2}-4X^{2}Y^{4}Z^{4}-5Y^{8}Z^{2}+2Y^{6}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.48.0-40.bl.1.7	$80$	$6$	$6$	$0$	$?$
80.144.3-40.bx.1.12	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$