Modular curve 80.288.5-40.gp.2.9

• Label: 80.288.5-40.gp.2.9
• 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:	$\begin{bmatrix}2&59\\55&46\end{bmatrix}$, $\begin{bmatrix}37&24\\56&5\end{bmatrix}$, $\begin{bmatrix}41&72\\34&39\end{bmatrix}$, $\begin{bmatrix}44&59\\1&62\end{bmatrix}$, $\begin{bmatrix}67&4\\68&43\end{bmatrix}$, $\begin{bmatrix}79&62\\76&25\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.5.gp.2 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 z + y^{2} $
	$=$	$5 x z + 5 y^{2} + w^{2} - w t$
	$=$	$10 x^{2} + 10 z^{2} + 2 w^{2} + 2 w t + t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 2 x^{4} z^{2} + 5 x^{2} y^{4} + 8 x^{2} y^{2} z^{2} + 4 x^{2} z^{4} + 2 y^{4} z^{2} $

Rational points

This modular curve has no real points, 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{460800z^{2}w^{14}t^{2}-1152000z^{2}w^{13}t^{3}+806400z^{2}w^{12}t^{4}-115200z^{2}w^{10}t^{6}+10598400z^{2}w^{9}t^{7}-13248000z^{2}w^{8}t^{8}-1324800z^{2}w^{7}t^{9}+3312000z^{2}w^{6}t^{10}+662400z^{2}w^{5}t^{11}+2620800z^{2}w^{4}t^{12}-1146600z^{2}w^{2}t^{14}-409500z^{2}wt^{15}-40950z^{2}t^{16}-4096w^{18}+12288w^{17}t+33792w^{16}t^{2}-65024w^{15}t^{3}-11520w^{14}t^{4}-694272w^{13}t^{5}+1284096w^{12}t^{6}+689664w^{11}t^{7}-450048w^{10}t^{8}-927360w^{9}t^{9}-2031168w^{8}t^{10}+1114944w^{7}t^{11}+1179696w^{6}t^{12}+393168w^{5}t^{13}+16380w^{4}t^{14}-286714w^{3}t^{15}-200703w^{2}t^{16}-49152wt^{17}-4096t^{18}}{t^{2}w^{5}(w-t)(640z^{2}w^{8}-960z^{2}w^{7}t+160z^{2}w^{6}t^{2}+160z^{2}w^{5}t^{3}+160z^{2}w^{3}t^{5}-40z^{2}w^{2}t^{6}-60z^{2}wt^{7}-10z^{2}t^{8}+64w^{10}-32w^{9}t-48w^{8}t^{2}+80w^{7}t^{3}-48w^{6}t^{4}+12w^{4}t^{6}-2w^{3}t^{7}-w^{2}t^{8})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.5.gp.2 :

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

Equation of the image curve:

$0$

$=$

$ 5X^{2}Y^{4}+2X^{4}Z^{2}+8X^{2}Y^{2}Z^{2}+2Y^{4}Z^{2}+4X^{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.1.23	$80$	$2$	$2$	$3$	$?$