Modular curve 80.288.5-40.gt.1.43

• Label: 80.288.5-40.gt.1.43
• Level: $80$
• Index: $288$
• Genus: $5$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$320$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps:	$16$ (none of which are rational)		Cusp widths	$1^{4}\cdot2^{2}\cdot5^{4}\cdot8^{2}\cdot10^{2}\cdot40^{2}$		Cusp orbits	$2^{8}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$2 \le \gamma \le 8$
$\overline{\Q}$-gonality:	$2 \le \gamma \le 5$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40M5

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}9&28\\50&67\end{bmatrix}$, $\begin{bmatrix}44&45\\61&28\end{bmatrix}$, $\begin{bmatrix}52&41\\3&10\end{bmatrix}$, $\begin{bmatrix}60&79\\7&12\end{bmatrix}$, $\begin{bmatrix}60&79\\17&2\end{bmatrix}$, $\begin{bmatrix}64&25\\57&32\end{bmatrix}$, $\begin{bmatrix}66&71\\63&74\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.5.gt.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 $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $	$=$	$ x t + y^{2} $
	$=$	$ - x t + y^{2} - z w$
	$=$	$2 x^{2} - x t + y^{2} - 5 z^{2} + 3 z w - w^{2} + 2 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{4} y^{2} - 2 x^{4} z^{2} - 8 x^{2} y^{2} z^{2} - 2 y^{4} z^{2} + 20 y^{2} z^{4} $

Rational points

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

Maps to other modular curves

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

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2}\cdot\frac{37500xz^{16}t+453410000xz^{14}t^{3}+202626460000xz^{12}t^{5}+19659524328000xz^{10}t^{7}+827979230496000xz^{8}t^{9}+20407186530400000xz^{6}t^{11}+346402524531571200xz^{4}t^{13}+4470452794209356800xz^{2}t^{15}+245760xw^{16}t+80281600xw^{14}t^{3}+9433907200xw^{12}t^{5}+655235481600xw^{10}t^{7}+32924526182400xw^{8}t^{9}+1325370992230400xw^{6}t^{11}+45334624311705600xw^{4}t^{13}+1367799500374016000xw^{2}t^{15}+26831712313694745600xt^{17}+125z^{18}+9992250z^{16}t^{2}+18953300000z^{14}t^{4}+3227192824000z^{12}t^{6}+184461732196000z^{10}t^{8}+5491955086856000z^{8}t^{10}+105773620424140800z^{6}t^{12}+1492453208893696000z^{4}t^{14}+16676739653188972800z^{2}t^{16}+4096w^{18}+3317760w^{16}t^{2}+535429120w^{14}t^{4}+43598479360w^{12}t^{6}+2407523287040w^{10}t^{8}+103121023139840w^{8}t^{10}+3684919683317760w^{6}t^{12}+114829712529817600w^{4}t^{14}+3212936169101721600w^{2}t^{16}-6916400382593728000t^{18}}{tz^{4}(z^{2}-2t^{2})^{3}(5z^{2}-2t^{2})(5xz^{4}+20xz^{2}t^{2}+4xt^{4}+20z^{4}t-8z^{2}t^{3})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.5.gt.1 :

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle t$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{2}z$

Equation of the image curve:

$0$

$=$

$ X^{4}Y^{2}-2X^{4}Z^{2}-8X^{2}Y^{2}Z^{2}-2Y^{4}Z^{2}+20Y^{2}Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.21	$80$	$2$	$2$	$3$	$?$