Modular curve 40.288.5-40.hw.1.7

• Label: 40.288.5-40.hw.1.7
• Level: $40$
• Index: $288$
• Genus: $5$
• Analytic rank: $0$
• Cusps: $16$
• $\Q$-cusps: $0$

Invariants

Level:	$40$		$\SL_2$-level:	$20$		Newform level:	$1600$
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	$2^{4}\cdot4^{4}\cdot10^{4}\cdot20^{4}$		Cusp orbits	$2^{6}\cdot4$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$4$
$\overline{\Q}$-gonality:	$4$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	20I5
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	40.288.5.170

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}25&6\\38&3\end{bmatrix}$, $\begin{bmatrix}33&1\\12&17\end{bmatrix}$, $\begin{bmatrix}33&31\\6&3\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.5.hw.1 for the level structure with $-I$)
Cyclic 40-isogeny field degree:	$4$
Cyclic 40-torsion field degree:	$32$
Full 40-torsion field degree:	$2560$

Jacobian

Conductor:	$2^{26}\cdot5^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	100.2.a.a, 320.2.a.d, 320.2.c.b, 1600.2.a.w

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $	$=$	$ x y + y^{2} - z^{2} $
	$=$	$2 x y - 5 x z + 2 y^{2} + 3 z^{2} - w^{2}$
	$=$	$5 x^{2} + 3 x y + 10 x z + 3 y^{2} + 7 z^{2} - 2 w^{2} + 2 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 2240 x^{8} + 320 x^{7} y - 8960 x^{7} z + 96 x^{6} y^{2} - 1920 x^{6} y z - 14720 x^{6} z^{2} + \cdots + 74065 z^{8} $

Rational points

This modular curve has no $\Q_p$ points for $p=3$, 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 2^6\,\frac{156240z^{2}w^{16}-350640z^{2}w^{14}t^{2}+258480z^{2}w^{12}t^{4}+258480z^{2}w^{10}t^{6}-1213200z^{2}w^{8}t^{8}+1735920z^{2}w^{6}t^{10}-1151820z^{2}w^{4}t^{12}+351540z^{2}w^{2}t^{14}-39060z^{2}t^{16}-6248w^{18}+10752w^{16}t^{2}-432w^{14}t^{4}-74860w^{12}t^{6}+282720w^{10}t^{8}-466944w^{8}t^{10}+392186w^{6}t^{12}-172656w^{4}t^{14}+37500w^{2}t^{16}-3125t^{18}}{t^{2}w^{4}(80z^{2}w^{10}+100z^{2}w^{8}t^{2}+100z^{2}w^{6}t^{4}+100z^{2}w^{4}t^{6}-200z^{2}w^{2}t^{8}+40z^{2}t^{10}-16w^{12}-12w^{10}t^{2}-9w^{8}t^{4}-8w^{6}t^{6}+4w^{4}t^{8})}$

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

$\displaystyle X$	$=$	$\displaystyle x-\frac{4}{5}w$
$\displaystyle Y$	$=$	$\displaystyle 4y+4t$
$\displaystyle Z$	$=$	$\displaystyle 2z+\frac{2}{5}w$

Equation of the image curve:

$0$

$=$

$ 2240X^{8}+320X^{7}Y+96X^{6}Y^{2}+8X^{5}Y^{3}+X^{4}Y^{4}-8960X^{7}Z-1920X^{6}YZ-768X^{5}Y^{2}Z-80X^{4}Y^{3}Z-12X^{3}Y^{4}Z-14720X^{6}Z^{2}+1280X^{5}YZ^{2}+1520X^{4}Y^{2}Z^{2}+280X^{3}Y^{3}Z^{2}+54X^{2}Y^{4}Z^{2}+75520X^{5}Z^{3}+9600X^{4}YZ^{3}+1840X^{3}Y^{2}Z^{3}-360X^{2}Y^{3}Z^{3}-108XY^{4}Z^{3}+30800X^{4}Z^{4}-13000X^{3}YZ^{4}-7570X^{2}Y^{2}Z^{4}+81Y^{4}Z^{4}-197920X^{3}Z^{5}-8160X^{2}YZ^{5}+2652XY^{2}Z^{5}+216Y^{3}Z^{5}-47520X^{2}Z^{6}+10160XYZ^{6}+4014Y^{2}Z^{6}+160560XZ^{7}+5160YZ^{7}+74065Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.144.1-20.m.2.7	$20$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-20.m.2.7	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-40.ba.1.3	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-40.ba.1.6	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.3-40.eq.2.9	$40$	$2$	$2$	$3$	$0$	$1^{2}$
40.144.3-40.eq.2.14	$40$	$2$	$2$	$3$	$0$	$1^{2}$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
40.1440.37-40.pr.1.2	$40$	$5$	$5$	$37$	$5$	$1^{16}\cdot2^{8}$