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Properties

Label 60.60.4.cj.1
Level $60$
Index $60$
Genus $4$
Analytic rank $1$
Cusps $3$
$\Q$-cusps $1$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $3600$
Index: $60$ $\PSL_2$-index:$60$
Genus: $4 = 1 + \frac{ 60 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$
Cusps: $3$ (of which $1$ is rational) Cusp widths $20^{3}$ Cusp orbits $1\cdot2$
Elliptic points: $2$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2 \le \gamma \le 4$
Rational cusps: $1$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 20C4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.60.4.46

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}3&44\\5&9\end{bmatrix}$, $\begin{bmatrix}13&14\\5&59\end{bmatrix}$, $\begin{bmatrix}33&28\\16&25\end{bmatrix}$, $\begin{bmatrix}49&6\\24&55\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $48$
Cyclic 60-torsion field degree: $768$
Full 60-torsion field degree: $36864$

Jacobian

Conductor: $2^{9}\cdot3^{4}\cdot5^{8}$
Simple: no
Squarefree: yes
Decomposition: $1^{4}$
Newforms: 50.2.a.b, 400.2.a.a, 450.2.a.g, 1800.2.a.r

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 90 x^{2} + 15 x y - x z + x w + 15 y^{2} - y z + y w + 2 z^{2} - 2 z w $
$=$ $30 x^{3} - x^{2} y - 3 x^{2} z + 3 x^{2} w - 2 x y^{2} + 4 x y w + 4 x z^{2} - 2 x z w - 2 x w^{2} + \cdots - 2 y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 11310 x^{6} - 8906 x^{5} y - 240 x^{5} z + 4111 x^{4} y^{2} - 137 x^{4} y z + 240 x^{4} z^{2} + \cdots + 30 y^{4} z^{2} $
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Rational points

This modular curve has 1 rational cusp and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.

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

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

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 -2^2\cdot3^6\,\frac{17161113600821886843485419350xyz^{9}+16187684328403919780088680325xyz^{8}w-43204352112996993868946354910xyz^{7}w^{2}-34592305609327169944385200080xyz^{6}w^{3}+22005900533638867294191933540xyz^{5}w^{4}-26664194908609532797416008625xyz^{4}w^{5}-27168818544938381446856157300xyz^{3}w^{6}+9637358052811019771677621860xyz^{2}w^{7}+6718961664097091673956938080xyzw^{8}+2169597862169073515627760xyw^{9}-14171247504675097660132798330xz^{10}-9281540978288277041848801505xz^{9}w+36965622198650940590523946773xz^{8}w^{2}+6340506564358168204055842206xz^{7}w^{3}-13994717762976059058048920616xz^{6}w^{4}+29680539285752176910455262247xz^{5}w^{5}+6521036262109340106566441865xz^{4}w^{6}-27151794731279022274667085588xz^{3}w^{7}-14675091756370927810156999596xz^{2}w^{8}-848254336341994235198571824xzw^{9}+511544232522819540651684368xw^{10}+2691162715769192515838868000y^{3}z^{8}+27270245025168104418814072125y^{3}z^{7}w+41229135680571350542870644825y^{3}z^{6}w^{2}+18415794346276118051607946425y^{3}z^{5}w^{3}+33921550284683501154957773625y^{3}z^{4}w^{4}+24829258352079968381402365500y^{3}z^{3}w^{5}-1085441800678914490183599750y^{3}z^{2}w^{6}-4336749007882765057944666450y^{3}zw^{7}-625847604550262452185279300y^{3}w^{8}-11938561604252458980092258700y^{2}z^{9}-40592203176176859572679955275y^{2}z^{8}w-24685064231167754210904142680y^{2}z^{7}w^{2}-6043668262946650269718142940y^{2}z^{6}w^{3}-10638830373672097618656639480y^{2}z^{5}w^{4}+2256790146483735995166966375y^{2}z^{4}w^{5}+22311971312576354144882839350y^{2}z^{3}w^{6}+14955728805480969310393029780y^{2}z^{2}w^{7}+1974591954577249763814206190y^{2}zw^{8}-283629719368622155955902620y^{2}w^{9}+5597728564403308712773852070yz^{10}+18623306268702475048536716920yz^{9}w-1514309007878872022957732292yz^{8}w^{2}-19569249080613207275545360314yz^{7}w^{3}-5106745003438984608815429706yz^{6}w^{4}-19735934258657808089624660628yz^{5}w^{5}-35030162508633599812104540210yz^{4}w^{6}-15195317725760183550413220348yz^{3}w^{7}-388679287496303764324814826yz^{2}w^{8}+652207626163620665751240466yzw^{9}+1543460624866378892698868yw^{10}+253656991935725650770144710z^{11}-281272701044808573641593640z^{10}w-885514174834083236350582026z^{9}w^{2}+1027281821832868841876719548z^{8}w^{3}+640502106295046057776313652z^{7}w^{4}-1176735167447539745377190244z^{6}w^{5}+260738610584113309211676120z^{5}w^{6}+636698918357886071278247256z^{4}w^{7}-289554564230351480631243018z^{3}w^{8}-167773729376258923885688372z^{2}w^{9}-1307251144707702821491486zw^{10}-171789265971480805312500w^{11}}{49215170596350083172973625550xyz^{9}+51744778601640756143991732225xyz^{8}w-637649615313551724005818980xyz^{7}w^{2}+18801509940327867097518810135xyz^{6}w^{3}+15725865072629641016730693870xyz^{5}w^{4}-15535826701874950555288629000xyz^{4}w^{5}-13932223020018568238559224400xyz^{3}w^{6}-2535860261299277952635263920xyz^{2}w^{7}+266703690730164641902038240xyzw^{8}+83717624525958120075161280xyw^{9}-52780434786152861206648081490xz^{10}-45701214971427545096184349765xz^{9}w+18213388882550024383173559719xz^{8}w^{2}+9399739451636527038117231543xz^{7}w^{3}+5979872659228573355191682727xz^{6}w^{4}+33035377728431845680671741466xz^{5}w^{5}+26745991991392738210003275720xz^{4}w^{6}+6255335892662016274877154336xz^{3}w^{7}-561366287417490918069455088xz^{2}w^{8}-377455110048048099182032672xzw^{9}-34750437323018143788226496xw^{10}-20586873740563855200445652250y^{3}z^{8}-11106807808215553096430521125y^{3}z^{7}w-14597382930341054127061493025y^{3}z^{6}w^{2}-18904764914890956383982717600y^{3}z^{5}w^{3}+281758793845213565833767750y^{3}z^{4}w^{4}+8115549029779728877058934000y^{3}z^{3}w^{5}+3300120620631174545469432000y^{3}z^{2}w^{6}+445473406765401584974034400y^{3}zw^{7}+26681437258690073692809600y^{3}w^{8}-35115260358946551502196729850y^{2}z^{9}-66644098388853070800785104575y^{2}z^{8}w-49556853420340158936698263290y^{2}z^{7}w^{2}-41800261235977166108544851445y^{2}z^{6}w^{3}-46288278426593424955903799940y^{2}z^{5}w^{4}-28624108487189128005432766500y^{2}z^{4}w^{5}-7272755415237448459427083200y^{2}z^{3}w^{6}-28134586745148847700414160y^{2}z^{2}w^{7}+236607369281500368485344320y^{2}zw^{8}+16429826212610412266168640y^{2}w^{9}+27287231760967776279202458460yz^{10}+73343763444505187397438170510yz^{9}w+66802730610816501025915709274yz^{8}w^{2}+50986703892903812854120746108yz^{7}w^{3}+50982092379290105965336019082yz^{6}w^{4}+29219208454060398377497969716yz^{5}w^{5}+5406541261465397419296378870yz^{4}w^{6}-996583841431608388037302944yz^{3}w^{7}-452712091836703432003488528yz^{2}w^{8}-42900680430890454920827552yzw^{9}-284224317839207456770496yw^{10}+1118142448618872296239065880z^{11}-270717553067652584522635420z^{10}w-1057720216286750857884760128z^{9}w^{2}-170289604975871986866556956z^{8}w^{3}-68059031028218961563853444z^{7}w^{4}-129325506256601252783736432z^{6}w^{5}+254824388693276733697051860z^{5}w^{6}+238779491805778765429493568z^{4}w^{7}+52841978491014666234139296z^{3}w^{8}+3028596940986813370750784z^{2}w^{9}+568448635678414913540992zw^{10}}$

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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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
20.30.2.l.1 $20$ $2$ $2$ $2$ $1$ $1^{2}$
60.12.0.bn.1 $60$ $5$ $5$ $0$ $0$ 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
60.120.8.e.1 $60$ $2$ $2$ $8$ $1$ $1^{4}$
60.120.8.m.1 $60$ $2$ $2$ $8$ $2$ $1^{4}$
60.120.8.z.1 $60$ $2$ $2$ $8$ $2$ $1^{4}$
60.120.8.bc.1 $60$ $2$ $2$ $8$ $3$ $1^{4}$
60.120.8.bv.1 $60$ $2$ $2$ $8$ $2$ $1^{4}$
60.120.8.bw.1 $60$ $2$ $2$ $8$ $3$ $1^{4}$
60.120.8.bz.1 $60$ $2$ $2$ $8$ $1$ $1^{4}$
60.120.8.cc.1 $60$ $2$ $2$ $8$ $2$ $1^{4}$
60.180.10.gb.1 $60$ $3$ $3$ $10$ $4$ $1^{6}$
60.180.13.nr.1 $60$ $3$ $3$ $13$ $7$ $1^{9}$
60.240.15.jj.1 $60$ $4$ $4$ $15$ $5$ $1^{11}$
60.240.18.v.1 $60$ $4$ $4$ $18$ $5$ $1^{14}$
120.120.8.bw.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.cp.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.hn.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.ij.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.kf.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.kn.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.lr.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.mj.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.nd.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.nf.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.pf.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.pl.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.pp.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.pr.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.ql.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.qr.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.rl.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.rn.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.sh.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.sn.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.tt.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.tv.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.tz.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.uf.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.9.cv.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.db.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.df.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.dh.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.qr.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.qx.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.rr.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.rt.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.xp.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.xv.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.yp.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.yr.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.yv.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.zb.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.bbb.1 $120$ $2$ $2$ $9$ $?$ not computed
120.120.9.bbd.1 $120$ $2$ $2$ $9$ $?$ not computed