Modular curve 40.288.5-40.es.1.7

• Label: 40.288.5-40.es.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:	40.288.5.1286

Level structure

$\GL_2(\Z/40\Z)$-generators:	$\begin{bmatrix}19&2\\38&13\end{bmatrix}$, $\begin{bmatrix}21&30\\28&3\end{bmatrix}$, $\begin{bmatrix}25&29\\8&1\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.5.es.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:	$2^{24}\cdot5^{7}$
Simple:	no
Squarefree:	yes
Decomposition:	$1^{3}\cdot2$
Newforms:	80.2.c.a, 320.2.a.f, 400.2.a.e, 1600.2.a.w

Models

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

$ 0 $	$=$	$ x y - y^{2} + z^{2} $
	$=$	$x y + 2 x z - y^{2} - z^{2} + t^{2}$
	$=$	$2 x^{2} - x y + 4 x z + y^{2} + 3 z^{2} + 5 w^{2} - 2 t^{2}$

Singular plane model Singular plane model

$ 0 $

$=$

$ 14 x^{8} - 20 x^{7} y + 45 x^{6} y^{2} - 256 x^{6} z^{2} - 50 x^{5} y^{3} + 280 x^{5} y z^{2} + \cdots + 4400 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 $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle -\frac{1}{5}\cdot\frac{12206250000z^{2}w^{16}-43942500000z^{2}w^{14}t^{2}+57591000000z^{2}w^{12}t^{4}-34718400000z^{2}w^{10}t^{6}+9705600000z^{2}w^{8}t^{8}-827136000z^{2}w^{6}t^{10}-330854400z^{2}w^{4}t^{12}+179527680z^{2}w^{2}t^{14}-31997952z^{2}t^{16}+6103515625w^{18}-29296875000w^{16}t^{2}+53955000000w^{14}t^{4}-49023250000w^{12}t^{6}+23347200000w^{10}t^{8}-5654400000w^{8}t^{10}+598880000w^{6}t^{12}+1382400w^{4}t^{14}-13762560w^{2}t^{16}+3198976t^{18}}{t^{4}w^{2}(6250z^{2}w^{10}-12500z^{2}w^{8}t^{2}+2500z^{2}w^{6}t^{4}+1000z^{2}w^{4}t^{6}+400z^{2}w^{2}t^{8}+128z^{2}t^{10}+625w^{8}t^{4}-500w^{6}t^{6}-225w^{4}t^{8}-120w^{2}t^{10}-64t^{12})}$

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

$\displaystyle X$	$=$	$\displaystyle x-z$
$\displaystyle Y$	$=$	$\displaystyle y+w$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{2}t$

Equation of the image curve:

$0$

$=$

$ 14X^{8}-20X^{7}Y+45X^{6}Y^{2}-50X^{5}Y^{3}+25X^{4}Y^{4}-256X^{6}Z^{2}+280X^{5}YZ^{2}-340X^{4}Y^{2}Z^{2}+100X^{3}Y^{3}Z^{2}+1536X^{4}Z^{4}-680X^{3}YZ^{4}+300X^{2}Y^{2}Z^{4}-4160X^{2}Z^{6}+400XYZ^{6}+4400Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.144.3-20.bi.2.7	$20$	$2$	$2$	$3$	$1$	$1^{2}$
40.144.1-40.n.2.10	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-40.n.2.13	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-40.ba.1.3	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.1-40.ba.1.10	$40$	$2$	$2$	$1$	$0$	$1^{2}\cdot2$
40.144.3-20.bi.2.7	$40$	$2$	$2$	$3$	$1$	$1^{2}$

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.kj.1.2	$40$	$5$	$5$	$37$	$7$	$1^{16}\cdot2^{8}$