Invariants
| Level: | $20$ | $\SL_2$-level: | $20$ | Newform level: | $100$ | ||
| Index: | $60$ | $\PSL_2$-index: | $60$ | ||||
| Genus: | $3 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $5^{4}\cdot20^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20E3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 20.60.3.1 |
Level structure
Jacobian
| Conductor: | $2^{4}\cdot5^{6}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 50.2.a.b$^{2}$, 100.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ 2 x^{4} - 4 x^{3} y + 4 x^{3} z + 6 x^{2} y^{2} + 17 x^{2} y z + 5 x^{2} z^{2} - 4 x y^{3} - 17 x y^{2} z + \cdots - 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 to other modular curves
$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\cdot5^2\,\frac{2808x^{2}y^{13}+51880x^{2}y^{12}z+435274x^{2}y^{11}z^{2}+2365338x^{2}y^{10}z^{3}+9110190x^{2}y^{9}z^{4}+26111038x^{2}y^{8}z^{5}+56462820x^{2}y^{7}z^{6}+91565124x^{2}y^{6}z^{7}+109022548x^{2}y^{5}z^{8}+90187620x^{2}y^{4}z^{9}+48965042x^{2}y^{3}z^{10}+16482690x^{2}y^{2}z^{11}+3136806x^{2}yz^{12}+265302x^{2}z^{13}-2808xy^{14}-49072xy^{13}z-383394xy^{12}z^{2}-1930064xy^{11}z^{3}-6744852xy^{10}z^{4}-17000848xy^{9}z^{5}-30351782xy^{8}z^{6}-35102304xy^{7}z^{7}-17457424xy^{6}z^{8}+18834928xy^{5}z^{9}+41222578xy^{4}z^{10}+32482352xy^{3}z^{11}+13345884xy^{2}z^{12}+2871504xyz^{13}+265302xz^{14}+1728y^{15}+15952y^{14}z+90460y^{13}z^{2}+336519y^{12}z^{3}+884310y^{11}z^{4}+1667224y^{10}z^{5}+2104542y^{9}z^{6}+1573455y^{8}z^{7}-543716y^{7}z^{8}-3474120y^{6}z^{9}-7892872y^{5}z^{10}-11484267y^{4}z^{11}-9655250y^{3}z^{12}-4597344y^{2}z^{13}-1170450yz^{14}-132651z^{15}}{7x^{2}y^{13}+593x^{2}y^{12}z+11914x^{2}y^{11}z^{2}+92914x^{2}y^{10}z^{3}+353505x^{2}y^{9}z^{4}+731187x^{2}y^{8}z^{5}+850168x^{2}y^{7}z^{6}+549044x^{2}y^{6}z^{7}+196209x^{2}y^{5}z^{8}+45715x^{2}y^{4}z^{9}+6598x^{2}y^{3}z^{10}+642x^{2}y^{2}z^{11}+31x^{2}yz^{12}+x^{2}z^{13}-7xy^{14}-586xy^{13}z-11321xy^{12}z^{2}-81000xy^{11}z^{3}-260591xy^{10}z^{4}-377682xy^{9}z^{5}-118981xy^{8}z^{6}+301124xy^{7}z^{7}+352835xy^{6}z^{8}+150494xy^{5}z^{9}+39117xy^{4}z^{10}+5956xy^{3}z^{11}+611xy^{2}z^{12}+30xyz^{13}+xz^{14}+6y^{15}+341y^{14}z+3925y^{13}z^{2}+14416y^{12}z^{3}+19374y^{11}z^{4}+7451y^{10}z^{5}+4921y^{9}z^{6}-8160y^{8}z^{7}-73144y^{7}z^{8}-100901y^{6}z^{9}-52451y^{5}z^{10}-14656y^{4}z^{11}-2580y^{3}z^{12}-251y^{2}z^{13}-19yz^{14}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(5)$ | $5$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(4)$ | $4$ | $10$ | $10$ | $0$ | $0$ | full Jacobian |
| 10.30.1.a.1 | $10$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 20.120.5.i.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.120.5.j.1 | $20$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 20.120.5.m.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.120.5.n.1 | $20$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 20.120.7.b.1 | $20$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 20.120.7.p.1 | $20$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 20.120.7.s.1 | $20$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 20.120.7.t.1 | $20$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 20.120.7.w.1 | $20$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 20.120.7.x.1 | $20$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 20.120.7.ba.1 | $20$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 20.120.7.bb.1 | $20$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 20.180.7.m.1 | $20$ | $3$ | $3$ | $7$ | $0$ | $1^{4}$ |
| 40.120.5.be.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.120.5.bh.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.120.5.bq.1 | $40$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 40.120.5.bt.1 | $40$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 40.120.7.j.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.120.7.bu.1 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.120.7.cc.1 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 40.120.7.cf.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.ci.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.120.7.cj.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.120.7.ck.1 | $40$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 40.120.7.cl.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.120.7.cm.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.120.7.cn.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.co.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.120.7.cp.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.cq.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.cr.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.120.7.cs.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.ct.1 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.120.7.cu.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.cv.1 | $40$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 40.120.7.cw.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.120.7.cx.1 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 40.120.7.de.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.120.7.dh.1 | $40$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 40.120.7.dq.1 | $40$ | $2$ | $2$ | $7$ | $0$ | $1^{4}$ |
| 40.120.7.dt.1 | $40$ | $2$ | $2$ | $7$ | $4$ | $1^{4}$ |
| 60.120.5.i.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.120.5.j.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.120.5.m.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{2}$ |
| 60.120.5.n.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}$ |
| 60.120.7.s.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.120.7.t.1 | $60$ | $2$ | $2$ | $7$ | $2$ | $1^{4}$ |
| 60.120.7.w.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.120.7.x.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.120.7.ba.1 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 60.120.7.bb.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.120.7.be.1 | $60$ | $2$ | $2$ | $7$ | $3$ | $1^{4}$ |
| 60.120.7.bf.1 | $60$ | $2$ | $2$ | $7$ | $1$ | $1^{4}$ |
| 60.180.13.be.1 | $60$ | $3$ | $3$ | $13$ | $5$ | $1^{10}$ |
| 60.240.15.bq.1 | $60$ | $4$ | $4$ | $15$ | $0$ | $1^{12}$ |
| 100.300.23.c.1 | $100$ | $5$ | $5$ | $23$ | $?$ | not computed |
| 120.120.5.be.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.bh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.bq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.5.bt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.120.7.cc.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cf.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.co.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cr.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cu.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cv.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cw.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cx.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cy.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.cz.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.da.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.db.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dc.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dd.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.de.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.df.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dg.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dh.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.di.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dj.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dq.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.dt.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.ec.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 120.120.7.ef.1 | $120$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.5.i.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.j.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.m.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.5.n.1 | $140$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 140.120.7.s.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.t.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.w.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.x.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.ba.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.bb.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.be.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 140.120.7.bf.1 | $140$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.5.i.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.j.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.m.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.5.n.1 | $220$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 220.120.7.s.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.t.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.w.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.x.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.ba.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.bb.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.be.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 220.120.7.bf.1 | $220$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.5.i.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.j.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.m.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.5.n.1 | $260$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 260.120.7.s.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.t.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.w.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.x.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.ba.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.bb.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.be.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 260.120.7.bf.1 | $260$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.5.be.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.bh.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.bq.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.5.bt.1 | $280$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 280.120.7.cc.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cf.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.co.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cr.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cu.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cv.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cw.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cx.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cy.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.cz.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.da.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.db.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dc.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dd.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.de.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.df.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dg.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dh.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.di.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dj.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dq.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.dt.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.ec.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 280.120.7.ef.1 | $280$ | $2$ | $2$ | $7$ | $?$ | not computed |