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Properties

Label 60.72.1.fa.1
Level $60$
Index $72$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $24$
Index: $72$ $\PSL_2$-index:$72$
Genus: $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $6^{4}\cdot12^{4}$ Cusp orbits $2^{4}$
Elliptic points: $8$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12T1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.72.1.131

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}11&30\\27&17\end{bmatrix}$, $\begin{bmatrix}27&14\\43&3\end{bmatrix}$, $\begin{bmatrix}37&6\\51&17\end{bmatrix}$, $\begin{bmatrix}49&30\\9&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $24$
Cyclic 60-torsion field degree: $384$
Full 60-torsion field degree: $30720$

Jacobian

Conductor: $2^{3}\cdot3$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 24.2.a.a

Models

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

$ 0 $ $=$ $ 8 x y + x z - x w + 4 y^{2} + y z + z^{2} - w^{2} $
$=$ $7 x^{2} + 7 x y - x z - x w - 4 y^{2} - y z - z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 644 x^{4} - 8 x^{3} y + 97 x^{3} z - x^{2} y^{2} - 2 x^{2} y z + 81 x^{2} z^{2} - 2 x y z^{2} + \cdots - 16 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle 15z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^{12}}\cdot\frac{409314473926598008469723886140626487216882880xz^{17}+425616562005975396017972556727421407976352576xz^{16}w+854826921619644409791776310294025396263348480xz^{15}w^{2}+824450408069896584192128652223256622066349824xz^{14}w^{3}+1053296956666140054459070681480205190145803264xz^{13}w^{4}+953845105026949142847645059793852641920134144xz^{12}w^{5}+907700780881776510738319955824046666567522304xz^{11}w^{6}+769807999455980641647434955881636580595574784xz^{10}w^{7}+650894732086643393509800418289228853398357760xz^{9}w^{8}+473468495160724768967879886269255395206508800xz^{8}w^{9}+325174335421179425523328960150717642140222336xz^{7}w^{10}+213166044043493330884428489963222127185975936xz^{6}w^{11}+143296084941664819026447985035675612553582896xz^{5}w^{12}+85002270592899063873070360431194883755473296xz^{4}w^{13}+38945743490188662191201380468140333577135836xz^{3}w^{14}+19465942731549746049549944400214971344502180xz^{2}w^{15}+7465107224921845086736420391282740291668909xzw^{16}+6441290590480930485829023527495636379051795xw^{17}+601126393452385430281521057486703423277131008y^{2}z^{16}+505043314576885225326518438794013784915265536y^{2}z^{15}w+1085322130177689962421123787515804416980884480y^{2}z^{14}w^{2}+836576073639134400876472945844173083137900544y^{2}z^{13}w^{3}+1073060250031971291652432511096820955492823040y^{2}z^{12}w^{4}+916641142364021528548463959835141936607461376y^{2}z^{11}w^{5}+896718707421440966438086882971453823017246720y^{2}z^{10}w^{6}+704758853504058546252573913986521665543618560y^{2}z^{9}w^{7}+559426918793664662779471811522321910588103680y^{2}z^{8}w^{8}+364600617842796232542897556727448153040650240y^{2}z^{7}w^{9}+260517363010374945901564269247858187949780480y^{2}z^{6}w^{10}+172083688104519708068292284850119584875869184y^{2}z^{5}w^{11}+106640674834474461722768362571995676351367360y^{2}z^{4}w^{12}+47307131420194052475628815231813154556130816y^{2}z^{3}w^{13}+25342184544167141799756646617647211863671920y^{2}z^{2}w^{14}+9000153611066504785655066230737009170574624y^{2}zw^{15}+9456482681322913448169458797217376816929172y^{2}w^{16}+125893992341391343545166776825212620338607296yz^{17}+126260828644221306331629609698503446228816384yz^{16}w+433500756352840254498743137149281646238111488yz^{15}w^{2}+356575320281806128629888370357513636689965056yz^{14}w^{3}+587629344236386138797651312464134786767725568yz^{13}w^{4}+442696930332750907748297540486123143231881216yz^{12}w^{5}+482230556134640897072864396451455553836636160yz^{11}w^{6}+413131827736043395312770601022837136523530240yz^{10}w^{7}+377635806267399022060093149306354311441544960yz^{9}w^{8}+261551871472241490944084224298561017800990720yz^{8}w^{9}+183979403185986608581568055736250884247112576yz^{7}w^{10}+120598308808846727151799111868076390665354496yz^{6}w^{11}+87981779575210184069655128765967636094984752yz^{5}w^{12}+51260702570081749504829670226860151617807744yz^{4}w^{13}+24531293240249571531560189255851143946375452yz^{3}w^{14}+10580603520783838996917081495868486428354888yz^{2}w^{15}+5215068822155218890322653833598487999025253yzw^{16}+4458226264239342967930619571104780229205056yw^{17}+145277565687842840192502710484129285281608384z^{18}+126260828644221306331629609698503446228816384z^{17}w+294252574896134339521429892851987129919963712z^{16}w^{2}+276041928961859687929714369284438497608487424z^{15}w^{3}+427585376995158797696751445920370650876156672z^{14}w^{4}+373332855053383976281184775382516661615763456z^{13}w^{5}+395167777163567067108964744829029614234731520z^{12}w^{6}+324800403451424236022796451469594527874371584z^{11}w^{7}+295201669019369292335444091748497232247676672z^{10}w^{8}+228004744922576672721049216705681612322877440z^{9}w^{9}+175425966529046364195756420892504935547724928z^{8}w^{10}+112669528078947427095121708712440624396160256z^{7}w^{11}+74152398685894248566415223231395825398869680z^{6}w^{12}+48416168731506205215699729462251714468793984z^{5}w^{13}+29853099713210730704085690520991102999318508z^{4}w^{14}+12963674673918815464210778596207504966504776z^{3}w^{15}+6698738592777171451385589104880476605453833z^{2}w^{16}+2250038402766626196413766557684252292643656zw^{17}+2285932659779892153013027919811429039581131w^{18}}{9578791428565442945009385600000xz^{17}+330780591238154505734983593600000xz^{16}w+5172364359801219351439662089184000xz^{15}w^{2}+50086966334333719555494241010668800xz^{14}w^{3}+321219084261695090307381076706865600xz^{13}w^{4}+1522598766003084836391390791279746752xz^{12}w^{5}+5131127742595753573048669334019492288xz^{11}w^{6}+13627912892584453200895714146581610432xz^{10}w^{7}+26025149945733809809117485717447563988xz^{9}w^{8}+40543857803192819661544349062363129668xz^{8}w^{9}+46669346172478027501910698981575152493xz^{7}w^{10}+45346819873844911124721521376584859771xz^{6}w^{11}+37011530876734150903098441235497222144xz^{5}w^{12}+21870587628574003393605267789402341376xz^{4}w^{13}+8624291076498497047211289223785086976xz^{3}w^{14}+2049533798705599009695031007539888128xz^{2}w^{15}+274894686104274358321017236823736320xzw^{16}+15473014879780384082065626130022400xw^{17}+13374731696236433942258188800000y^{2}z^{16}+451900189041837079797878169600000y^{2}z^{15}w+7188385859142378536138162567040000y^{2}z^{14}w^{2}+67116880737701458292504555585740800y^{2}z^{13}w^{3}+443329901074229923907531300558227200y^{2}z^{12}w^{4}+1978679362871350286939952581848390656y^{2}z^{11}w^{5}+6979442253526295054708611670643899136y^{2}z^{10}w^{6}+16853695950567262039259188552894656000y^{2}z^{9}w^{7}+33959117122028241539798472950578164048y^{2}z^{8}w^{8}+46248331182385144673578468124819525760y^{2}z^{7}w^{9}+53477194947556226951756752417468925172y^{2}z^{6}w^{10}+45014928166809245635972607179009032192y^{2}z^{5}w^{11}+28581531049944792698026262212509696000y^{2}z^{4}w^{12}+11322665521442072552199891647333400576y^{2}z^{3}w^{13}+2812282590580294826878331622134906880y^{2}z^{2}w^{14}+374071191451620786152024252381921280y^{2}zw^{15}+21604752968290690634663282815795200y^{2}w^{16}+3343682924059108485564547200000yz^{17}+112975047260459269949469542400000yz^{16}w+1800421855987998012412911222240000yz^{15}w^{2}+16894820869215848369015143573555200yz^{14}w^{3}+112612906104969532173115376910897600yz^{13}w^{4}+512122957739360208635738553403832064yz^{12}w^{5}+1853962750105147497327343620622352064yz^{11}w^{6}+4741437018621314406349782902181748608yz^{10}w^{7}+10193877148113475196853943849071510612yz^{9}w^{8}+16228599083239081975774214076852957216yz^{8}w^{9}+21655932803093734976762165546488084797yz^{7}w^{10}+24506730956557941979489606350777711168yz^{6}w^{11}+22795060801411037436522767336010940416yz^{5}w^{12}+14023648156480688434767999663807135744yz^{4}w^{13}+5697798616774314932939460234640883712yz^{3}w^{14}+1341091510187081190216301539112255488yz^{2}w^{15}+181376888241369161783011173728256000yzw^{16}+10071826637707711423399805426073600yw^{17}+3343682924059108485564547200000z^{18}+112975047260459269949469542400000z^{17}w+1798150724781132752528170923360000z^{16}w^{2}+16816095410934003257212137445939200z^{15}w^{3}+111413454046917932783703949939896000z^{14}w^{4}+500411002040073105339673770986168064z^{13}w^{5}+1782206350987095924755362627523371264z^{12}w^{6}+4396224149782792139508039984842250624z^{11}w^{7}+9124818810806633908250903232026670996z^{10}w^{8}+13334333861651915568096812602229529504z^{9}w^{9}+16991724694546111860508545480115889577z^{8}w^{10}+17288525458306440456333431431159556208z^{7}w^{11}+15363480090725661194981196956475116403z^{6}w^{12}+12294849993551743505612430681405652992z^{5}w^{13}+7381472059084736156056414763293605888z^{4}w^{14}+2861573638842087026909190210419949568z^{3}w^{15}+704773637079572913086613934304133120z^{2}w^{16}+93517797862905196538006063095480320zw^{17}+5401188242072672658665820703948800w^{18}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.36.1.bo.1 $12$ $2$ $2$ $1$ $0$ dimension zero
30.36.0.d.1 $30$ $2$ $2$ $0$ $0$ full Jacobian
60.36.0.f.1 $60$ $2$ $2$ $0$ $0$ 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
60.144.5.j.1 $60$ $2$ $2$ $5$ $1$ $1^{4}$
60.144.5.bz.1 $60$ $2$ $2$ $5$ $1$ $1^{4}$
60.144.5.jw.1 $60$ $2$ $2$ $5$ $0$ $1^{4}$
60.144.5.jz.1 $60$ $2$ $2$ $5$ $0$ $1^{4}$
60.144.5.mp.1 $60$ $2$ $2$ $5$ $0$ $1^{4}$
60.144.5.mr.1 $60$ $2$ $2$ $5$ $0$ $1^{4}$
60.144.5.mw.1 $60$ $2$ $2$ $5$ $1$ $1^{4}$
60.144.5.mz.1 $60$ $2$ $2$ $5$ $1$ $1^{4}$
60.360.25.cfp.1 $60$ $5$ $5$ $25$ $6$ $1^{24}$
60.432.25.bkj.1 $60$ $6$ $6$ $25$ $2$ $1^{24}$
60.720.49.ejz.1 $60$ $10$ $10$ $49$ $9$ $1^{48}$
120.144.5.mv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.nz.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.cyo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.czj.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dqz.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.drn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dtd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.dtr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.fsr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.fst.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.hgr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.hgt.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ibo.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.ibr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.iwn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.5.iwp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.144.9.hkf.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.hkh.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.ldc.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.ldf.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.oev.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.oex.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.qpr.1 $120$ $2$ $2$ $9$ $?$ not computed
120.144.9.qpt.1 $120$ $2$ $2$ $9$ $?$ not computed
180.216.9.bj.1 $180$ $3$ $3$ $9$ $?$ not computed
180.216.9.bv.1 $180$ $3$ $3$ $9$ $?$ not computed
180.216.9.cu.1 $180$ $3$ $3$ $9$ $?$ not computed