Properties

Label 20.72.3.bt.2
Level $20$
Index $72$
Genus $3$
Analytic rank $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $20$ $\SL_2$-level: $20$ Newform level: $80$
Index: $72$ $\PSL_2$-index:$72$
Genus: $3 = 1 + \frac{ 72 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $4^{3}\cdot20^{3}$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $4$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 3$
$\overline{\Q}$-gonality: $2 \le \gamma \le 3$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 20K3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.72.3.70

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}1&6\\11&15\end{bmatrix}$, $\begin{bmatrix}13&8\\15&19\end{bmatrix}$, $\begin{bmatrix}13&12\\10&17\end{bmatrix}$, $\begin{bmatrix}17&8\\2&17\end{bmatrix}$
$\GL_2(\Z/20\Z)$-subgroup: $D_{20}:C_4^2$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 20-isogeny field degree: $2$
Cyclic 20-torsion field degree: $16$
Full 20-torsion field degree: $640$

Jacobian

Conductor: $2^{12}\cdot5^{3}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 80.2.a.a, 80.2.c.a

Models

Canonical model in $\mathbb{P}^{ 2 }$

$ 0 $ $=$ $ 4 x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + 2 x y^{3} + 2 x y z^{2} - y^{2} z^{2} - z^{4} $
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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.

Canonical model
$(0:1:0)$, $(-1:1:0)$

Maps to other modular curves

$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^6\,\frac{480x^{3}y^{15}-4832x^{3}y^{13}z^{2}+21024x^{3}y^{11}z^{4}-27808x^{3}y^{9}z^{6}-2016x^{3}y^{7}z^{8}+13536x^{3}y^{5}z^{10}-2592x^{3}y^{3}z^{12}-864x^{3}yz^{14}+4x^{2}y^{16}-1584x^{2}y^{14}z^{2}+4664x^{2}y^{12}z^{4}-5760x^{2}y^{10}z^{6}-7248x^{2}y^{8}z^{8}+7632x^{2}y^{6}z^{10}+3816x^{2}y^{4}z^{12}-6048x^{2}y^{2}z^{14}+1836x^{2}z^{16}-352xy^{17}+4000xy^{15}z^{2}-11320xy^{13}z^{4}+11280xy^{11}z^{6}+7320xy^{9}z^{8}-16320xy^{7}z^{10}+6840xy^{5}z^{12}+1296xy^{3}z^{14}-1080xyz^{16}-y^{18}+167y^{16}z^{2}-1952y^{14}z^{4}+4972y^{12}z^{6}-4242y^{10}z^{8}-5862y^{8}z^{10}+7192y^{6}z^{12}-612y^{4}z^{14}-1701y^{2}z^{16}+567z^{18}}{(y^{2}+z^{2})^{5}(16x^{3}y^{5}-32x^{3}y^{3}z^{2}-32x^{3}yz^{4}+2x^{2}y^{6}+6x^{2}y^{4}z^{2}+68x^{2}z^{6}-14xy^{7}+30xy^{5}z^{2}-28xy^{3}z^{4}-40xyz^{6}+7y^{6}z^{2}-12y^{4}z^{4}+6y^{2}z^{6}+21z^{8})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
20.36.1.j.1 $20$ $2$ $2$ $1$ $0$ $2$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
20.144.5.bh.2 $20$ $2$ $2$ $5$ $0$ $1^{2}$
20.144.5.bj.1 $20$ $2$ $2$ $5$ $0$ $1^{2}$
20.144.5.bk.2 $20$ $2$ $2$ $5$ $0$ $1^{2}$
20.144.5.bm.1 $20$ $2$ $2$ $5$ $1$ $1^{2}$
20.144.7.i.2 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.144.7.r.1 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.144.7.z.1 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.144.7.bc.2 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.144.7.bd.2 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.144.7.bf.1 $20$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
20.144.7.bi.2 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.144.7.bk.1 $20$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
20.360.21.f.1 $20$ $5$ $5$ $21$ $2$ $1^{8}\cdot2^{5}$
40.144.5.kp.2 $40$ $2$ $2$ $5$ $1$ $1^{2}$
40.144.5.kt.1 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.144.5.lh.2 $40$ $2$ $2$ $5$ $1$ $1^{2}$
40.144.5.ll.1 $40$ $2$ $2$ $5$ $2$ $1^{2}$
40.144.7.cg.1 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.dk.1 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.ht.1 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.ik.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.jn.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.jr.1 $40$ $2$ $2$ $7$ $2$ $1^{2}\cdot2$
40.144.7.kj.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.kn.1 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.7.kz.1 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.7.lb.1 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.lh.2 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.7.lj.2 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.7.mt.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.mz.2 $40$ $2$ $2$ $7$ $2$ $1^{2}\cdot2$
40.144.7.nb.1 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.7.nh.1 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.oj.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.ol.2 $40$ $2$ $2$ $7$ $2$ $1^{2}\cdot2$
40.144.7.op.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.ov.2 $40$ $2$ $2$ $7$ $2$ $1^{2}\cdot2$
40.144.7.ql.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.qn.2 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.7.qr.2 $40$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
40.144.7.qx.2 $40$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
40.144.9.eb.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.144.9.eh.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.144.9.el.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.144.9.en.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.144.9.ib.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.144.9.ih.2 $40$ $2$ $2$ $9$ $2$ $1^{2}\cdot2^{2}$
40.144.9.il.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot2^{2}$
40.144.9.in.2 $40$ $2$ $2$ $9$ $2$ $1^{2}\cdot2^{2}$
40.144.9.kn.2 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
40.144.9.kt.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
40.144.9.kv.1 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
40.144.9.lb.1 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
40.144.9.ml.1 $40$ $2$ $2$ $9$ $2$ $1^{2}\cdot4$
40.144.9.mn.1 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
40.144.9.mt.2 $40$ $2$ $2$ $9$ $1$ $1^{2}\cdot4$
40.144.9.mv.2 $40$ $2$ $2$ $9$ $0$ $1^{2}\cdot4$
60.144.5.rx.2 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.144.5.sb.1 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.144.5.se.2 $60$ $2$ $2$ $5$ $0$ $1^{2}$
60.144.5.si.1 $60$ $2$ $2$ $5$ $1$ $1^{2}$
60.144.7.kg.2 $60$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
60.144.7.km.1 $60$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
60.144.7.mt.2 $60$ $2$ $2$ $7$ $2$ $1^{2}\cdot2$
60.144.7.mz.2 $60$ $2$ $2$ $7$ $2$ $1^{2}\cdot2$
60.144.7.to.2 $60$ $2$ $2$ $7$ $0$ $1^{2}\cdot2$
60.144.7.ts.1 $60$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
60.144.7.tx.2 $60$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
60.144.7.ub.1 $60$ $2$ $2$ $7$ $1$ $1^{2}\cdot2$
60.216.13.sa.2 $60$ $3$ $3$ $13$ $3$ $1^{4}\cdot2^{3}$
60.288.19.r.2 $60$ $4$ $4$ $19$ $1$ $1^{8}\cdot2^{4}$
100.360.21.f.2 $100$ $5$ $5$ $21$ $?$ not computed
120.144.5.fbn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.fbv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.fcz.2 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.fdh.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.7.gug.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.gvs.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.hqf.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.hrn.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.laj.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lar.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lbz.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lch.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.ldj.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.ldl.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.ldz.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.leb.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lgr.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lgx.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lhh.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lhn.1 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lmh.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lmj.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lmn.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lmt.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lrv.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lrx.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lsb.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.7.lsh.2 $120$ $2$ $2$ $7$ $?$ not computed
120.144.9.obh.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.obn.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.obr.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.obt.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.svn.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.svt.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.svx.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.svz.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgpj.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgpp.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgpz.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgqf.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgsv.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgsx.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgtl.2 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.bgtn.2 $120$ $2$ $2$ $9$ $?$ not computed
140.144.5.iv.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.ix.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.jd.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.5.jf.1 $140$ $2$ $2$ $5$ $?$ not computed
140.144.7.cm.2 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.cs.1 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.cy.1 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.de.2 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.dk.2 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.dm.1 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.ds.2 $140$ $2$ $2$ $7$ $?$ not computed
140.144.7.du.1 $140$ $2$ $2$ $7$ $?$ not computed
220.144.5.iv.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.ix.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.jd.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.5.jf.2 $220$ $2$ $2$ $5$ $?$ not computed
220.144.7.cd.2 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.cj.1 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.cp.1 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.cv.2 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.db.2 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.dd.1 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.dj.2 $220$ $2$ $2$ $7$ $?$ not computed
220.144.7.dl.1 $220$ $2$ $2$ $7$ $?$ not computed
260.144.5.iv.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.ix.2 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.jd.1 $260$ $2$ $2$ $5$ $?$ not computed
260.144.5.jf.2 $260$ $2$ $2$ $5$ $?$ not computed
260.144.7.do.1 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.du.1 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.ea.2 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.eg.2 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.em.1 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.eo.2 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.eu.1 $260$ $2$ $2$ $7$ $?$ not computed
260.144.7.ew.2 $260$ $2$ $2$ $7$ $?$ not computed
280.144.5.cnr.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.cnv.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.cpf.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.5.cpj.1 $280$ $2$ $2$ $5$ $?$ not computed
280.144.7.ul.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.vp.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.xr.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.yv.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bbh.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bbl.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bcv.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bcz.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bfh.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bfj.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bfp.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bfr.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bhb.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bhh.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bhj.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bhp.1 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bkn.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bkp.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bkt.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bkz.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bof.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.boh.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bol.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.7.bor.2 $280$ $2$ $2$ $7$ $?$ not computed
280.144.9.xd.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.xj.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.xn.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.xp.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bav.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bbb.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bbf.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bbh.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bef.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bel.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.ben.1 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bet.1 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bgd.1 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bgf.1 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bgl.2 $280$ $2$ $2$ $9$ $?$ not computed
280.144.9.bgn.2 $280$ $2$ $2$ $9$ $?$ not computed