Modular curve 40.288.5-40.ii.1.7

• Label: 40.288.5-40.ii.1.7
• Level: $40$
• Index: $288$
• Genus: $5$
• Analytic rank: $1$
• 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	$2^{4}\cdot4^{2}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$1$
$\Q$-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:	40.288.5.191

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}15&16\\26&25\end{bmatrix}$, $\begin{bmatrix}37&37\\26&33\end{bmatrix}$, $\begin{bmatrix}37&37\\30&19\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.5.ii.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:	$2^{28}\cdot5^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	320.2.a.d, 320.2.c.b, 400.2.a.c, 1600.2.a.c

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 $

$=$

$ 960 x^{8} - 320 x^{7} y - 3840 x^{7} z - 64 x^{6} y^{2} + 320 x^{6} y z + 24320 x^{6} z^{2} + \cdots + 119185 z^{8} $

Rational points

This modular curve has no $\Q_p$ points for $p=19$, 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.ii.1 :

$\displaystyle X$	$=$	$\displaystyle x+\frac{6}{5}w$
$\displaystyle Y$	$=$	$\displaystyle 4y+4t$
$\displaystyle Z$	$=$	$\displaystyle 2z+\frac{2}{5}w$

Equation of the image curve:

$0$

$=$

$ 960X^{8}-320X^{7}Y-64X^{6}Y^{2}+8X^{5}Y^{3}+X^{4}Y^{4}-3840X^{7}Z+320X^{6}YZ-128X^{5}Y^{2}Z+40X^{4}Y^{3}Z+8X^{3}Y^{4}Z+24320X^{6}Z^{2}-2080X^{5}YZ^{2}-400X^{4}Y^{2}Z^{2}+60X^{3}Y^{3}Z^{2}+24X^{2}Y^{4}Z^{2}-59520X^{5}Z^{3}-1600X^{4}YZ^{3}-1520X^{3}Y^{2}Z^{3}+40X^{2}Y^{3}Z^{3}+32XY^{4}Z^{3}+165200X^{4}Z^{4}+2600X^{3}YZ^{4}-1050X^{2}Y^{2}Z^{4}+80XY^{3}Z^{4}+16Y^{4}Z^{4}-235680X^{3}Z^{5}-13240X^{2}YZ^{5}-808XY^{2}Z^{5}+96Y^{3}Z^{5}+311920X^{2}Z^{6}+9340XYZ^{6}-3176Y^{2}Z^{6}-203360XZ^{7}-9960YZ^{7}+119185Z^{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.k.2.7	$20$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-20.k.2.3	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-40.bf.1.3	$40$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
40.144.1-40.bf.1.12	$40$	$2$	$2$	$1$	$1$	$1^{2}\cdot2$
40.144.3-40.eq.2.14	$40$	$2$	$2$	$3$	$0$	$1^{2}$
40.144.3-40.eq.2.15	$40$	$2$	$2$	$3$	$0$	$1^{2}$

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.pk.1.1	$40$	$5$	$5$	$37$	$7$	$1^{16}\cdot2^{8}$