Modular curve 20.24.1.e.1

• Label: 20.24.1.e.1
• Level: $20$
• Index: $24$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $4$
• $\Q$-cusps: $2$

Invariants

Level:	$20$		$\SL_2$-level:	$10$		Newform level:	$400$
Index:	$24$		$\PSL_2$-index:	$24$
Genus:	$1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps:	$4$ (of which $2$ are rational)		Cusp widths	$2^{2}\cdot10^{2}$		Cusp orbits	$1^{2}\cdot2$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	$0$
$\Q$-gonality:	$2$
$\overline{\Q}$-gonality:	$2$
Rational cusps:	$2$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:	10D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	20.24.1.4

Level structure

$\GL_2(\Z/20\Z)$-generators:	$\begin{bmatrix}3&18\\13&15\end{bmatrix}$, $\begin{bmatrix}12&11\\15&19\end{bmatrix}$, $\begin{bmatrix}18&19\\3&12\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	20.48.1-20.e.1.1, 20.48.1-20.e.1.2, 20.48.1-20.e.1.3, 20.48.1-20.e.1.4, 40.48.1-20.e.1.1, 40.48.1-20.e.1.2, 40.48.1-20.e.1.3, 40.48.1-20.e.1.4, 60.48.1-20.e.1.1, 60.48.1-20.e.1.2, 60.48.1-20.e.1.3, 60.48.1-20.e.1.4, 120.48.1-20.e.1.1, 120.48.1-20.e.1.2, 120.48.1-20.e.1.3, 120.48.1-20.e.1.4, 140.48.1-20.e.1.1, 140.48.1-20.e.1.2, 140.48.1-20.e.1.3, 140.48.1-20.e.1.4, 220.48.1-20.e.1.1, 220.48.1-20.e.1.2, 220.48.1-20.e.1.3, 220.48.1-20.e.1.4, 260.48.1-20.e.1.1, 260.48.1-20.e.1.2, 260.48.1-20.e.1.3, 260.48.1-20.e.1.4, 280.48.1-20.e.1.1, 280.48.1-20.e.1.2, 280.48.1-20.e.1.3, 280.48.1-20.e.1.4
Cyclic 20-isogeny field degree:	$6$
Cyclic 20-torsion field degree:	$24$
Full 20-torsion field degree:	$1920$

Jacobian

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

Models

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ x^{3} + x^{2} - 1033x + 12438 $

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model

$(18:0:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{5^5}\cdot\frac{40x^{2}y^{6}-975000x^{2}y^{4}z^{2}+7364375000x^{2}y^{2}z^{4}-18316640625000x^{2}z^{6}-1840xy^{6}z+37950000xy^{4}z^{3}-274286250000xy^{2}z^{5}+667657656250000xz^{7}-y^{8}+37660y^{6}z^{2}-498793750y^{4}z^{4}+2881368437500y^{2}z^{6}-6083246494140625z^{8}}{z^{3}y^{2}(3050x^{2}z+xy^{2}-111175xz^{2}-73y^{2}z+1012950z^{3})}$

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
$X_{\pm1}(5)$	$5$	$2$	$2$	$0$	$0$	full Jacobian
20.12.0.p.2	$20$	$2$	$2$	$0$	$0$	full Jacobian
20.12.1.b.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
20.72.1.i.1	$20$	$3$	$3$	$1$	$0$	dimension zero
20.96.5.g.2	$20$	$4$	$4$	$5$	$0$	$1^{2}\cdot2$
20.120.5.u.1	$20$	$5$	$5$	$5$	$0$	$1^{2}\cdot2$
60.72.5.cl.2	$60$	$3$	$3$	$5$	$0$	$1^{2}\cdot2$
60.96.5.bp.2	$60$	$4$	$4$	$5$	$0$	$1^{2}\cdot2$
100.120.5.e.1	$100$	$5$	$5$	$5$	$?$	not computed
140.192.13.h.2	$140$	$8$	$8$	$13$	$?$	not computed
220.288.21.h.2	$220$	$12$	$12$	$21$	$?$	not computed