Modular curve 20.36.1.b.1

• Label: 20.36.1.b.1
• Level: $20$
• Index: $36$
• Genus: $1$
• Analytic rank: $0$
• Cusps: $6$
• $\Q$-cusps: $2$

Invariants

Level:	$20$		$\SL_2$-level:	$20$		Newform level:	$80$
Index:	$36$		$\PSL_2$-index:	$36$
Genus:	$1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps:	$6$ (of which $2$ are rational)		Cusp widths	$1^{2}\cdot4\cdot5^{2}\cdot20$		Cusp orbits	$1^{2}\cdot2^{2}$
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:	20D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label:	20.36.1.3

Level structure

$\GL_2(\Z/20\Z)$-generators:	$\begin{bmatrix}1&16\\9&13\end{bmatrix}$, $\begin{bmatrix}7&4\\13&3\end{bmatrix}$, $\begin{bmatrix}11&6\\3&19\end{bmatrix}$, $\begin{bmatrix}17&0\\3&9\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup:	$D_{10}:C_4^3$
Contains $-I$:	yes
Quadratic refinements:	40.72.1-20.b.1.1, 40.72.1-20.b.1.2, 40.72.1-20.b.1.3, 40.72.1-20.b.1.4, 40.72.1-20.b.1.5, 40.72.1-20.b.1.6, 40.72.1-20.b.1.7, 40.72.1-20.b.1.8, 40.72.1-20.b.1.9, 40.72.1-20.b.1.10, 40.72.1-20.b.1.11, 40.72.1-20.b.1.12, 40.72.1-20.b.1.13, 40.72.1-20.b.1.14, 40.72.1-20.b.1.15, 40.72.1-20.b.1.16, 120.72.1-20.b.1.1, 120.72.1-20.b.1.2, 120.72.1-20.b.1.3, 120.72.1-20.b.1.4, 120.72.1-20.b.1.5, 120.72.1-20.b.1.6, 120.72.1-20.b.1.7, 120.72.1-20.b.1.8, 120.72.1-20.b.1.9, 120.72.1-20.b.1.10, 120.72.1-20.b.1.11, 120.72.1-20.b.1.12, 120.72.1-20.b.1.13, 120.72.1-20.b.1.14, 120.72.1-20.b.1.15, 120.72.1-20.b.1.16, 280.72.1-20.b.1.1, 280.72.1-20.b.1.2, 280.72.1-20.b.1.3, 280.72.1-20.b.1.4, 280.72.1-20.b.1.5, 280.72.1-20.b.1.6, 280.72.1-20.b.1.7, 280.72.1-20.b.1.8, 280.72.1-20.b.1.9, 280.72.1-20.b.1.10, 280.72.1-20.b.1.11, 280.72.1-20.b.1.12, 280.72.1-20.b.1.13, 280.72.1-20.b.1.14, 280.72.1-20.b.1.15, 280.72.1-20.b.1.16
Cyclic 20-isogeny field degree:	$2$
Cyclic 20-torsion field degree:	$16$
Full 20-torsion field degree:	$1280$

Jacobian

Conductor:	$2^{4}\cdot5$
Simple:	yes
Squarefree:	yes
Decomposition:	$1$
Newforms:	80.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $

$=$

$ x^{3} - x^{2} + 4x - 4 $

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

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

Maps to other modular curves

$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{18x^{2}y^{10}+231870x^{2}y^{8}z^{2}+160411560x^{2}y^{6}z^{4}+2836033434x^{2}y^{4}z^{6}+5360733242x^{2}y^{2}z^{8}+5211922921x^{2}z^{10}+819xy^{10}z+2126268xy^{8}z^{3}+512947965xy^{6}z^{5}-122871624xy^{4}z^{7}+6960980885xy^{2}z^{9}-7562391552xz^{11}+y^{12}+10983y^{10}z^{2}+23771328y^{8}z^{4}+1733869315y^{6}z^{6}+10433358193y^{4}z^{8}+23727088765y^{2}z^{10}+32868046756z^{12}}{z(50x^{2}y^{8}z+2918x^{2}y^{6}z^{3}-104248x^{2}y^{4}z^{5}-958112x^{2}y^{2}z^{7}-515968x^{2}z^{9}+xy^{10}-320xy^{8}z^{2}+15859xy^{6}z^{4}+333620xy^{4}z^{6}+39632xy^{2}z^{8}-4376384xz^{10}+15y^{10}z-1205y^{8}z^{3}-33613y^{6}z^{5}+80052y^{4}z^{7}+2000144y^{2}z^{9}+4892352z^{11})}$

Modular covers

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
4.6.0.b.1	$4$	$6$	$6$	$0$	$0$	full Jacobian
$X_0(5)$	$5$	$6$	$6$	$0$	$0$	full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank	Kernel decomposition
4.6.0.b.1	$4$	$6$	$6$	$0$	$0$	full Jacobian
$X_0(10)$	$10$	$2$	$2$	$0$	$0$	full Jacobian

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.d.1	$20$	$2$	$2$	$1$	$0$	dimension zero
20.72.1.d.2	$20$	$2$	$2$	$1$	$0$	dimension zero
20.72.1.e.1	$20$	$2$	$2$	$1$	$0$	dimension zero
20.72.1.e.2	$20$	$2$	$2$	$1$	$0$	dimension zero
20.72.3.a.1	$20$	$2$	$2$	$3$	$0$	$1^{2}$
20.72.3.g.1	$20$	$2$	$2$	$3$	$0$	$1^{2}$
20.72.3.m.1	$20$	$2$	$2$	$3$	$1$	$1^{2}$
20.72.3.n.1	$20$	$2$	$2$	$3$	$0$	$1^{2}$
20.72.3.q.1	$20$	$2$	$2$	$3$	$0$	$2$
20.72.3.q.2	$20$	$2$	$2$	$3$	$0$	$2$
20.72.3.r.1	$20$	$2$	$2$	$3$	$0$	$2$
20.72.3.r.2	$20$	$2$	$2$	$3$	$0$	$2$
20.180.7.g.1	$20$	$5$	$5$	$7$	$1$	$1^{6}$
40.72.1.m.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.1.m.2	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.1.p.1	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.1.p.2	$40$	$2$	$2$	$1$	$0$	dimension zero
40.72.3.g.1	$40$	$2$	$2$	$3$	$2$	$1^{2}$
40.72.3.t.1	$40$	$2$	$2$	$3$	$0$	$1^{2}$
40.72.3.bk.1	$40$	$2$	$2$	$3$	$1$	$1^{2}$
40.72.3.bn.1	$40$	$2$	$2$	$3$	$1$	$1^{2}$
40.72.3.ci.1	$40$	$2$	$2$	$3$	$0$	$2$
40.72.3.ci.2	$40$	$2$	$2$	$3$	$0$	$2$
40.72.3.cl.1	$40$	$2$	$2$	$3$	$0$	$2$
40.72.3.cl.2	$40$	$2$	$2$	$3$	$0$	$2$
60.72.1.k.1	$60$	$2$	$2$	$1$	$0$	dimension zero
60.72.1.k.2	$60$	$2$	$2$	$1$	$0$	dimension zero
60.72.1.l.1	$60$	$2$	$2$	$1$	$0$	dimension zero
60.72.1.l.2	$60$	$2$	$2$	$1$	$0$	dimension zero
60.72.3.eq.1	$60$	$2$	$2$	$3$	$0$	$1^{2}$
60.72.3.er.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.3.fc.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.3.fd.1	$60$	$2$	$2$	$3$	$1$	$1^{2}$
60.72.3.hs.1	$60$	$2$	$2$	$3$	$0$	$2$
60.72.3.hs.2	$60$	$2$	$2$	$3$	$0$	$2$
60.72.3.ht.1	$60$	$2$	$2$	$3$	$0$	$2$
60.72.3.ht.2	$60$	$2$	$2$	$3$	$0$	$2$
60.108.7.b.1	$60$	$3$	$3$	$7$	$1$	$1^{6}$
60.144.7.gw.1	$60$	$4$	$4$	$7$	$0$	$1^{6}$
100.180.7.b.1	$100$	$5$	$5$	$7$	$?$	not computed
120.72.1.bk.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.bk.2	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.bn.1	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.1.bn.2	$120$	$2$	$2$	$1$	$?$	dimension zero
120.72.3.bdw.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.bdz.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.bgm.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.bgp.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.cge.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.cge.2	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.cgh.1	$120$	$2$	$2$	$3$	$?$	not computed
120.72.3.cgh.2	$120$	$2$	$2$	$3$	$?$	not computed
140.72.1.d.1	$140$	$2$	$2$	$1$	$?$	dimension zero
140.72.1.d.2	$140$	$2$	$2$	$1$	$?$	dimension zero
140.72.1.e.1	$140$	$2$	$2$	$1$	$?$	dimension zero
140.72.1.e.2	$140$	$2$	$2$	$1$	$?$	dimension zero
140.72.3.m.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.n.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.q.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.r.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.u.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.u.2	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.v.1	$140$	$2$	$2$	$3$	$?$	not computed
140.72.3.v.2	$140$	$2$	$2$	$3$	$?$	not computed
140.288.19.b.1	$140$	$8$	$8$	$19$	$?$	not computed
220.72.1.d.1	$220$	$2$	$2$	$1$	$?$	dimension zero
220.72.1.d.2	$220$	$2$	$2$	$1$	$?$	dimension zero
220.72.1.e.1	$220$	$2$	$2$	$1$	$?$	dimension zero
220.72.1.e.2	$220$	$2$	$2$	$1$	$?$	dimension zero
220.72.3.m.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.n.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.q.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.r.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.u.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.u.2	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.v.1	$220$	$2$	$2$	$3$	$?$	not computed
220.72.3.v.2	$220$	$2$	$2$	$3$	$?$	not computed
260.72.1.d.1	$260$	$2$	$2$	$1$	$?$	dimension zero
260.72.1.d.2	$260$	$2$	$2$	$1$	$?$	dimension zero
260.72.1.e.1	$260$	$2$	$2$	$1$	$?$	dimension zero
260.72.1.e.2	$260$	$2$	$2$	$1$	$?$	dimension zero
260.72.3.m.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.n.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.q.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.r.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.u.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.u.2	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.v.1	$260$	$2$	$2$	$3$	$?$	not computed
260.72.3.v.2	$260$	$2$	$2$	$3$	$?$	not computed
280.72.1.m.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.1.m.2	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.1.p.1	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.1.p.2	$280$	$2$	$2$	$1$	$?$	dimension zero
280.72.3.bk.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.bn.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.bw.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.bz.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.cu.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.cu.2	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.cx.1	$280$	$2$	$2$	$3$	$?$	not computed
280.72.3.cx.2	$280$	$2$	$2$	$3$	$?$	not computed