Modular curve 20.40.1.c.1

• Label: 20.40.1.c.1
• Level: $20$
• Index: $40$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $4$
• $\Q$-cusps: $0$

Invariants

Level:	$20$		$\SL_2$-level:	$10$		Newform level:	$400$
Index:	$40$		$\PSL_2$-index:	$40$
Genus:	$1 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (none of which are rational)		Cusp widths	$10^{4}$		Cusp orbits	$4$
Elliptic points:	$0$ of order $2$ and $4$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$0$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10H1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	20.40.1.8

Level structure

$\GL_2(\Z/20\Z)$-generators:	$\begin{bmatrix}3&17\\17&14\end{bmatrix}$, $\begin{bmatrix}5&4\\1&15\end{bmatrix}$, $\begin{bmatrix}12&15\\5&17\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	20.80.1-20.c.1.1, 20.80.1-20.c.1.2, 40.80.1-20.c.1.1, 40.80.1-20.c.1.2, 60.80.1-20.c.1.1, 60.80.1-20.c.1.2, 120.80.1-20.c.1.1, 120.80.1-20.c.1.2, 140.80.1-20.c.1.1, 140.80.1-20.c.1.2, 220.80.1-20.c.1.1, 220.80.1-20.c.1.2, 260.80.1-20.c.1.1, 260.80.1-20.c.1.2, 280.80.1-20.c.1.1, 280.80.1-20.c.1.2
Cyclic 20-isogeny field degree:	$36$
Cyclic 20-torsion field degree:	$288$
Full 20-torsion field degree:	$1152$

Jacobian

Conductor:	$2^{4}\cdot5^{2}$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	400.2.a.c

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

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

Singular plane model Singular plane model

$ 0 $

$=$

$ 5 x^{4} - 10 x^{2} y^{2} - 5 x^{2} z^{2} + 9 y^{4} + 6 y^{2} z^{2} + z^{4} $

Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle x$
$\displaystyle Z$	$=$	$\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle 5^2\,\frac{14413248yz^{9}-71724128yz^{7}w^{2}-137511444yz^{5}w^{4}+462498456yz^{3}w^{6}+127541120yzw^{8}+929664z^{10}-64983664z^{8}w^{2}+224903728z^{6}w^{4}-32154927z^{4}w^{6}-275717120z^{2}w^{8}-15366400w^{10}}{208525yz^{9}-353150yz^{7}w^{2}+248675yz^{5}w^{4}-85750yz^{3}w^{6}+12005yzw^{8}+13450z^{10}-84075z^{8}w^{2}+78400z^{6}w^{4}-15925z^{4}w^{6}-6860z^{2}w^{8}+2401w^{10}}$

Modular covers

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This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
$X_{\mathrm{ns}}^+(10)$	$10$	$2$	$2$	$0$	$0$	full Jacobian
20.20.0.a.1	$20$	$2$	$2$	$0$	$0$	full Jacobian
20.20.1.a.1	$20$	$2$	$2$	$1$	$0$	dimension zero

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

Covering curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
20.120.3.a.1	$20$	$3$	$3$	$3$	$0$	$1^{2}$
20.120.5.f.1	$20$	$3$	$3$	$5$	$1$	$1^{4}$
20.120.5.t.1	$20$	$3$	$3$	$5$	$0$	$1^{4}$
20.160.9.d.1	$20$	$4$	$4$	$9$	$4$	$1^{8}$
60.120.9.i.1	$60$	$3$	$3$	$9$	$5$	$1^{6}\cdot2$
60.160.9.c.1	$60$	$4$	$4$	$9$	$2$	$1^{8}$
100.200.9.h.1	$100$	$5$	$5$	$9$	$?$	not computed
140.320.21.c.1	$140$	$8$	$8$	$21$	$?$	not computed