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Properties

Label 20.36.1.c.1
Level $20$
Index $36$
Genus $1$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $6$

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Invariants

Level: $20$ $\SL_2$-level: $20$ Newform level: $20$
Index: $36$ $\PSL_2$-index:$36$
Genus: $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (all of which are rational) Cusp widths $1^{2}\cdot4\cdot5^{2}\cdot20$ Cusp orbits $1^{6}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $6$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.36.1.1

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}3&12\\0&7\end{bmatrix}$, $\begin{bmatrix}7&1\\0&11\end{bmatrix}$, $\begin{bmatrix}11&10\\0&1\end{bmatrix}$, $\begin{bmatrix}13&5\\0&13\end{bmatrix}$, $\begin{bmatrix}13&16\\0&11\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup: $D_4\times C_{10}:C_4^2$
Contains $-I$: yes
Quadratic refinements: 20.72.1-20.c.1.1, 20.72.1-20.c.1.2, 20.72.1-20.c.1.3, 20.72.1-20.c.1.4, 20.72.1-20.c.1.5, 20.72.1-20.c.1.6, 20.72.1-20.c.1.7, 20.72.1-20.c.1.8, 40.72.1-20.c.1.1, 40.72.1-20.c.1.2, 40.72.1-20.c.1.3, 40.72.1-20.c.1.4, 40.72.1-20.c.1.5, 40.72.1-20.c.1.6, 40.72.1-20.c.1.7, 40.72.1-20.c.1.8, 40.72.1-20.c.1.9, 40.72.1-20.c.1.10, 40.72.1-20.c.1.11, 40.72.1-20.c.1.12, 40.72.1-20.c.1.13, 40.72.1-20.c.1.14, 40.72.1-20.c.1.15, 40.72.1-20.c.1.16, 40.72.1-20.c.1.17, 40.72.1-20.c.1.18, 40.72.1-20.c.1.19, 40.72.1-20.c.1.20, 40.72.1-20.c.1.21, 40.72.1-20.c.1.22, 40.72.1-20.c.1.23, 40.72.1-20.c.1.24, 60.72.1-20.c.1.1, 60.72.1-20.c.1.2, 60.72.1-20.c.1.3, 60.72.1-20.c.1.4, 60.72.1-20.c.1.5, 60.72.1-20.c.1.6, 60.72.1-20.c.1.7, 60.72.1-20.c.1.8, 120.72.1-20.c.1.1, 120.72.1-20.c.1.2, 120.72.1-20.c.1.3, 120.72.1-20.c.1.4, 120.72.1-20.c.1.5, 120.72.1-20.c.1.6, 120.72.1-20.c.1.7, 120.72.1-20.c.1.8, 120.72.1-20.c.1.9, 120.72.1-20.c.1.10, 120.72.1-20.c.1.11, 120.72.1-20.c.1.12, 120.72.1-20.c.1.13, 120.72.1-20.c.1.14, 120.72.1-20.c.1.15, 120.72.1-20.c.1.16, 120.72.1-20.c.1.17, 120.72.1-20.c.1.18, 120.72.1-20.c.1.19, 120.72.1-20.c.1.20, 120.72.1-20.c.1.21, 120.72.1-20.c.1.22, 120.72.1-20.c.1.23, 120.72.1-20.c.1.24, 140.72.1-20.c.1.1, 140.72.1-20.c.1.2, 140.72.1-20.c.1.3, 140.72.1-20.c.1.4, 140.72.1-20.c.1.5, 140.72.1-20.c.1.6, 140.72.1-20.c.1.7, 140.72.1-20.c.1.8, 220.72.1-20.c.1.1, 220.72.1-20.c.1.2, 220.72.1-20.c.1.3, 220.72.1-20.c.1.4, 220.72.1-20.c.1.5, 220.72.1-20.c.1.6, 220.72.1-20.c.1.7, 220.72.1-20.c.1.8, 260.72.1-20.c.1.1, 260.72.1-20.c.1.2, 260.72.1-20.c.1.3, 260.72.1-20.c.1.4, 260.72.1-20.c.1.5, 260.72.1-20.c.1.6, 260.72.1-20.c.1.7, 260.72.1-20.c.1.8, 280.72.1-20.c.1.1, 280.72.1-20.c.1.2, 280.72.1-20.c.1.3, 280.72.1-20.c.1.4, 280.72.1-20.c.1.5, 280.72.1-20.c.1.6, 280.72.1-20.c.1.7, 280.72.1-20.c.1.8, 280.72.1-20.c.1.9, 280.72.1-20.c.1.10, 280.72.1-20.c.1.11, 280.72.1-20.c.1.12, 280.72.1-20.c.1.13, 280.72.1-20.c.1.14, 280.72.1-20.c.1.15, 280.72.1-20.c.1.16, 280.72.1-20.c.1.17, 280.72.1-20.c.1.18, 280.72.1-20.c.1.19, 280.72.1-20.c.1.20, 280.72.1-20.c.1.21, 280.72.1-20.c.1.22, 280.72.1-20.c.1.23, 280.72.1-20.c.1.24
Cyclic 20-isogeny field degree: $1$
Cyclic 20-torsion field degree: $8$
Full 20-torsion field degree: $1280$

Jacobian

Conductor: $2^{2}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 20.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} + x^{2} + 4x + 4 $
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Rational points

This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(0:-2:1)$, $(-1:0:1)$, $(0:2:1)$, $(4:-10:1)$, $(0:1:0)$, $(4:10:1)$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{x^{2}y^{17}-1312x^{2}y^{15}z^{2}+151776x^{2}y^{14}z^{3}-3975880x^{2}y^{13}z^{4}+28829600x^{2}y^{12}z^{5}-247751440x^{2}y^{11}z^{6}-173455080x^{2}y^{10}z^{7}+28774520006x^{2}y^{9}z^{8}-12179629920x^{2}y^{8}z^{9}-149788365792x^{2}y^{7}z^{10}-2772139991360x^{2}y^{6}z^{11}+4965292102964x^{2}y^{5}z^{12}+31406365793072x^{2}y^{4}z^{13}-80531644416880x^{2}y^{3}z^{14}+4823382509736x^{2}y^{2}z^{15}+103593101229955x^{2}yz^{16}-57880589893632x^{2}z^{17}-26xy^{17}z+744xy^{16}z^{2}-54xy^{15}z^{3}-386880xy^{14}z^{4}+24484860xy^{13}z^{5}-343178480xy^{12}z^{6}+1737396299xy^{11}z^{7}-25473116640xy^{10}z^{8}+115282216130xy^{9}z^{9}-78595531080xy^{8}z^{10}+2651296456137xy^{7}z^{11}-14533740581632xy^{6}z^{12}+5648682909518xy^{5}z^{13}+40724091375240xy^{4}z^{14}+82422855106535xy^{3}z^{15}-395278188281856xy^{2}z^{16}+379841371176960xyz^{17}-91644267331584xz^{18}+272y^{17}z^{2}-17112y^{16}z^{3}+208510y^{15}z^{4}-441936y^{14}z^{5}-47936425y^{13}z^{6}+1709716400y^{12}z^{7}-9722411027y^{11}z^{8}+36842629584y^{10}z^{9}-424431181416y^{9}z^{10}+1246167350040y^{8}z^{11}+2144162763411y^{7}z^{12}-707096456984y^{6}z^{13}-54920889346691y^{5}z^{14}+92232176328248y^{4}z^{15}+128423460860505y^{3}z^{16}-404924953189728y^{2}z^{17}+276248269946380yz^{18}-33763677437952z^{19}}{z^{2}(204x^{2}y^{14}z+9860x^{2}y^{12}z^{3}+113535x^{2}y^{10}z^{5}-2520540x^{2}y^{8}z^{7}-12331080x^{2}y^{6}z^{9}+92274590x^{2}y^{4}z^{11}+16777241x^{2}y^{2}z^{13}-201326592x^{2}z^{15}+xy^{16}-520xy^{14}z^{2}+970xy^{12}z^{4}+689300xy^{10}z^{6}+2704835xy^{8}z^{8}-58510560xy^{6}z^{10}+10485765xy^{4}z^{12}+486539264xy^{2}z^{14}-318767104xz^{16}-23y^{16}z-594y^{14}z^{3}-70930y^{12}z^{5}-312958y^{10}z^{7}+10685375y^{8}z^{9}-18076735y^{6}z^{11}-165675405y^{4}z^{13}+452984932y^{2}z^{15}-117440512z^{17})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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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
$X_0(4)$ $4$ $6$ $6$ $0$ $0$ 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
$X_0(4)$ $4$ $6$ $6$ $0$ $0$ 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.f.1 $20$ $2$ $2$ $1$ $0$ dimension zero
20.72.1.f.2 $20$ $2$ $2$ $1$ $0$ dimension zero
20.72.1.g.1 $20$ $2$ $2$ $1$ $0$ dimension zero
20.72.1.g.2 $20$ $2$ $2$ $1$ $0$ dimension zero
20.72.3.b.1 $20$ $2$ $2$ $3$ $0$ $1^{2}$
20.72.3.l.1 $20$ $2$ $2$ $3$ $0$ $1^{2}$
20.72.3.o.1 $20$ $2$ $2$ $3$ $0$ $1^{2}$
20.72.3.p.1 $20$ $2$ $2$ $3$ $1$ $1^{2}$
20.72.3.s.1 $20$ $2$ $2$ $3$ $0$ $2$
20.72.3.s.2 $20$ $2$ $2$ $3$ $0$ $2$
20.72.3.t.1 $20$ $2$ $2$ $3$ $0$ $2$
20.72.3.t.2 $20$ $2$ $2$ $3$ $0$ $2$
20.180.7.i.1 $20$ $5$ $5$ $7$ $0$ $1^{6}$
40.72.1.s.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.72.1.s.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.72.1.v.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.72.1.v.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.72.3.h.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.72.3.bi.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.72.3.bq.1 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.72.3.bt.1 $40$ $2$ $2$ $3$ $2$ $1^{2}$
40.72.3.bw.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
$X_0(40)$ $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.72.3.by.1 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.72.3.bz.1 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.72.3.ca.1 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.ca.2 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cb.1 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cb.2 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cc.1 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cc.2 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cd.1 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cd.2 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.ce.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.72.3.cf.1 $40$ $2$ $2$ $3$ $2$ $1^{2}$
40.72.3.cg.1 $40$ $2$ $2$ $3$ $0$ $1^{2}$
40.72.3.ch.1 $40$ $2$ $2$ $3$ $1$ $1^{2}$
40.72.3.co.1 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.co.2 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cr.1 $40$ $2$ $2$ $3$ $0$ $2$
40.72.3.cr.2 $40$ $2$ $2$ $3$ $0$ $2$
60.72.1.m.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.72.1.m.2 $60$ $2$ $2$ $1$ $0$ dimension zero
60.72.1.n.1 $60$ $2$ $2$ $1$ $0$ dimension zero
60.72.1.n.2 $60$ $2$ $2$ $1$ $0$ dimension zero
60.72.3.es.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.72.3.et.1 $60$ $2$ $2$ $3$ $0$ $1^{2}$
60.72.3.fe.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.72.3.ff.1 $60$ $2$ $2$ $3$ $1$ $1^{2}$
60.72.3.hu.1 $60$ $2$ $2$ $3$ $0$ $2$
60.72.3.hu.2 $60$ $2$ $2$ $3$ $0$ $2$
60.72.3.hv.1 $60$ $2$ $2$ $3$ $0$ $2$
60.72.3.hv.2 $60$ $2$ $2$ $3$ $0$ $2$
60.108.7.c.1 $60$ $3$ $3$ $7$ $0$ $1^{6}$
$X_0(60)$ $60$ $4$ $4$ $7$ $0$ $1^{6}$
$X_0(100)$ $100$ $5$ $5$ $7$ $?$ not computed
120.72.1.bq.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.bq.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.bt.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.1.bt.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.72.3.bec.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.bef.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.bgs.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.bgv.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.bye.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byf.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byg.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byh.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byi.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byi.2 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byj.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byj.2 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byk.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byk.2 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byl.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byl.2 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.bym.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byo.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.byp.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.cgk.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.cgk.2 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.cgn.1 $120$ $2$ $2$ $3$ $?$ not computed
120.72.3.cgn.2 $120$ $2$ $2$ $3$ $?$ not computed
140.72.1.f.1 $140$ $2$ $2$ $1$ $?$ dimension zero
140.72.1.f.2 $140$ $2$ $2$ $1$ $?$ dimension zero
140.72.1.g.1 $140$ $2$ $2$ $1$ $?$ dimension zero
140.72.1.g.2 $140$ $2$ $2$ $1$ $?$ dimension zero
140.72.3.o.1 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.p.1 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.s.1 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.t.1 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.w.1 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.w.2 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.x.1 $140$ $2$ $2$ $3$ $?$ not computed
140.72.3.x.2 $140$ $2$ $2$ $3$ $?$ not computed
$X_0(140)$ $140$ $8$ $8$ $19$ $?$ not computed
220.72.1.f.1 $220$ $2$ $2$ $1$ $?$ dimension zero
220.72.1.f.2 $220$ $2$ $2$ $1$ $?$ dimension zero
220.72.1.g.1 $220$ $2$ $2$ $1$ $?$ dimension zero
220.72.1.g.2 $220$ $2$ $2$ $1$ $?$ dimension zero
220.72.3.o.1 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.p.1 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.s.1 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.t.1 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.w.1 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.w.2 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.x.1 $220$ $2$ $2$ $3$ $?$ not computed
220.72.3.x.2 $220$ $2$ $2$ $3$ $?$ not computed
260.72.1.f.1 $260$ $2$ $2$ $1$ $?$ dimension zero
260.72.1.f.2 $260$ $2$ $2$ $1$ $?$ dimension zero
260.72.1.g.1 $260$ $2$ $2$ $1$ $?$ dimension zero
260.72.1.g.2 $260$ $2$ $2$ $1$ $?$ dimension zero
260.72.3.o.1 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.p.1 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.s.1 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.t.1 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.w.1 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.w.2 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.x.1 $260$ $2$ $2$ $3$ $?$ not computed
260.72.3.x.2 $260$ $2$ $2$ $3$ $?$ not computed
280.72.1.s.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.72.1.s.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.72.1.v.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.72.1.v.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.72.3.bq.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.bt.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cc.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cf.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.ci.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cj.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.ck.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cl.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cm.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cm.2 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cn.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cn.2 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.co.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.co.2 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cp.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cp.2 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cq.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cr.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.cs.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.ct.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.da.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.da.2 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.dd.1 $280$ $2$ $2$ $3$ $?$ not computed
280.72.3.dd.2 $280$ $2$ $2$ $3$ $?$ not computed