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Properties

Label 60.192.1-60.f.2.14
Level $60$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $12$ Newform level: $600$
Index: $192$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ 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: 12V1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.192.1.288

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}31&20\\42&1\end{bmatrix}$, $\begin{bmatrix}37&36\\6&11\end{bmatrix}$, $\begin{bmatrix}43&56\\42&35\end{bmatrix}$, $\begin{bmatrix}53&38\\24&5\end{bmatrix}$
Contains $-I$: no $\quad$ (see 60.96.1.f.2 for the level structure with $-I$)
Cyclic 60-isogeny field degree: $12$
Cyclic 60-torsion field degree: $96$
Full 60-torsion field degree: $11520$

Jacobian

Conductor: $2^{3}\cdot3\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 600.2.a.h

Models

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

$ 0 $ $=$ $ 3 x y - 3 x z - 2 y^{2} + 2 y z + y w - 2 z^{2} + z w $
$=$ $3 x^{2} + 3 x y - 3 x z + 4 y^{2} + 2 y z + 3 y w + 4 z^{2} + 3 z w - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 11 x^{4} + x^{3} y - 22 x^{3} z - x^{2} y^{2} - 3 x^{2} y z + 24 x^{2} z^{2} + 4 x y^{2} z + \cdots + 11 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 to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3^6\cdot5^6\cdot11^2}\cdot\frac{3680071866199203189723108068217879168xz^{23}-50032313134568658479825494417489174848xz^{22}w-20479871917257787176534006223099341312xz^{21}w^{2}+35297275291767249114432850454828974752xz^{20}w^{3}+54116669229214542941824301442344689440xz^{19}w^{4}+23221075281906990475897695455373378576xz^{18}w^{5}-44948456177077773591786018849474700560xz^{17}w^{6}-7211442236386019469465876319758018816xz^{16}w^{7}-6869210426789911484776246808406999360xz^{15}w^{8}+18821533973800762058702796771247054080xz^{14}w^{9}-5129016226451685754479940678970320800xz^{13}w^{10}+543876613399156246767848608679945232xz^{12}w^{11}-2057997918417706196907896207762558424xz^{11}w^{12}+1207890530131345083133244724342339972xz^{10}w^{13}-79452995044843536121770087639240540xz^{9}w^{14}-111984808007833984624344983809218504xz^{8}w^{15}+34054105681278158475480122935004184xz^{7}w^{16}-2327740887624012570745432225023624xz^{6}w^{17}-453176153569472678227917488199636xz^{5}w^{18}+83319535587234250826978147270370xz^{4}w^{19}-3310563422863211693647892777838xz^{3}w^{20}-155031707852344384953311256315xz^{2}w^{21}+11002429372118493901210294803xzw^{22}-14203758212067115970229099172005362304y^{2}z^{22}-23659595462047500959573154605175340416y^{2}z^{21}w+76817073726781774852407202075937123232y^{2}z^{20}w^{2}+27959108572991999450920634747307558720y^{2}z^{19}w^{3}+29095163002833259154937465947013491040y^{2}z^{18}w^{4}-80442636212262805835008278251554635456y^{2}z^{17}w^{5}-2029136776078254154970923218033870768y^{2}z^{16}w^{6}-4393246045480113999694107758098871808y^{2}z^{15}w^{7}+29110000544042939210405494622197632960y^{2}z^{14}w^{8}-7426380548650903499340655224345754560y^{2}z^{13}w^{9}-1824398022351436224199862937334757136y^{2}z^{12}w^{10}-2003702594072214187239118843416736032y^{2}z^{11}w^{11}+1174141959892651834652776473805397616y^{2}z^{10}w^{12}+600602116033909581519316903507873440y^{2}z^{9}w^{13}-522648592625079082205979891037566660y^{2}z^{8}w^{14}+116499918290010518799305298669211632y^{2}z^{7}w^{15}+61317704259786670705423464212448y^{2}z^{6}w^{16}-3930135816315446959561481121934536y^{2}z^{5}w^{17}+574605537644167865962357478612310y^{2}z^{4}w^{18}-18368872336011627479846979997980y^{2}z^{3}w^{19}-1909945067754604969349653795722y^{2}z^{2}w^{20}+161368964124404577217750990444y^{2}zw^{21}-3666887163883707149036244101y^{2}w^{22}+182137093260718177584958181254414848yz^{23}+13787381091128483005113937117259696256yz^{22}w+16078650473441778743882871950414413824yz^{21}w^{2}+3522945720252003847329918605961344544yz^{20}w^{3}-27572315245945449528463595783134464960yz^{19}w^{4}-14957787356992725750772674494019752160yz^{18}w^{5}-5410493665639767748893273657668842176yz^{17}w^{6}+20184977516012573956936888114802445840yz^{16}w^{7}-1571338305777454719968132391489596928yz^{15}w^{8}+2496092857571162875050265278331592640yz^{14}w^{9}-6206908381991183203433567561004247872yz^{13}w^{10}+1992372503249012147210480978684623824yz^{12}w^{11}-175618139805341661337592872919247936yz^{11}w^{12}+546680084946781083298997564919672288yz^{10}w^{13}-379504144489203516403478077830941760yz^{9}w^{14}+67855566425319740083881206651443596yz^{8}w^{15}+12483222777114604421405271197535792yz^{7}w^{16}-6268762180147784853278561474593872yz^{6}w^{17}+770673188834746461020925270704448yz^{5}w^{18}-5805712766484786078997267544130yz^{4}w^{19}-5641452065435838816287169303180yz^{3}w^{20}+331595691061468455281595146618yz^{2}w^{21}+1178586978248302734375000yzw^{22}-333631480489116637838522191yw^{23}-13024774176078855312684103002257966400z^{24}-42194539425306839699664442477341340416z^{23}w+67855850225283294749706158664164914848z^{22}w^{2}+56846179421298569280217141823074288992z^{21}w^{3}+37070346021441868855117390824071044992z^{20}w^{4}-69612406607474025141271544663311852128z^{19}w^{5}-36872410336126251016563454640426790048z^{18}w^{6}+8869572654127978705267415117203830816z^{17}w^{7}+21893993033001486369084454207753090368z^{16}w^{8}+7986493038957753663124403321799177408z^{15}w^{9}-11915863456604385394596166964186103216z^{14}w^{10}+801157995191055990527537511909783888z^{13}w^{11}-392221246862936818843974292380666120z^{12}w^{12}+1825008728015993360823434608321646448z^{11}w^{13}-810285809764656322251451545775156872z^{10}w^{14}-2199693212398859616263120188412520z^{9}w^{15}+81965283380343638770526418876183288z^{8}w^{16}-20366276142446946742943122174373904z^{7}w^{17}+961708613170018911398723510351154z^{6}w^{18}+316717957642854581406960086698374z^{5}w^{19}-50215731770975942473346758437636z^{4}w^{20}+1894281918281349696020579599234z^{3}w^{21}+110690014109797710054381554912z^{2}w^{22}-11336060852607610539048816994zw^{23}+333425661488495661520162816w^{24}}{z^{2}(1051759920147788446373592xz^{21}-15527469929048089048817352xz^{20}w+32445792253168494806491380xz^{19}w^{2}+21887058453547859202815910xz^{18}w^{3}-118355188264853817026001780xz^{17}w^{4}+79080470914214924166570822xz^{16}w^{5}+70531627195877475317331708xz^{15}w^{6}-107843409761648649599522880xz^{14}w^{7}+25361129605790764995157080xz^{13}w^{8}+24915347791711496366911785xz^{12}w^{9}-16451663954447270446399755xz^{11}w^{10}+2573961370342186949339490xz^{10}w^{11}+546001917792460548375000xz^{9}w^{12}-251106060810726784265625xz^{8}w^{13}+39379541903085118453125xz^{7}w^{14}-3976374734828653265625xz^{6}w^{15}+294187333532236546875xz^{5}w^{16}-7997653335759656250xz^{4}w^{17}-913276014241593750xz^{3}w^{18}+108287930661328125xz^{2}w^{19}-3645267256171875xzw^{20}-4059416268506522588981976y^{2}z^{20}+4853481880823181562183920y^{2}z^{19}w+37054402826036112397713690y^{2}z^{18}w^{2}-87822834784145612029238580y^{2}z^{17}w^{3}+10949276671777839440381490y^{2}z^{16}w^{4}+138289613694265012390934688y^{2}z^{15}w^{5}-126912883856729644981568745y^{2}z^{14}w^{6}-15031656631243980174503760y^{2}z^{13}w^{7}+72640462136673267721445220y^{2}z^{12}w^{8}-32003515371253621084404570y^{2}z^{11}w^{9}-1533975688875668424139149y^{2}z^{10}w^{10}+4873629074485093213500000y^{2}z^{9}w^{11}-1513304657196412463796875y^{2}z^{8}w^{12}+239529563046284665343750y^{2}z^{7}w^{13}-31000175756043970781250y^{2}z^{6}w^{14}+4176235833072633000000y^{2}z^{5}w^{15}-429810473382979734375y^{2}z^{4}w^{16}+24616001286904500000y^{2}z^{3}w^{17}+105996387616968750y^{2}z^{2}w^{18}-106927839514375000y^{2}zw^{19}+4703177865562500y^{2}w^{20}+52054552636139691820512yz^{21}+3236804795117078488366272yz^{20}w-6174546186196581384304080yz^{19}w^{2}-7002708190047242343472230yz^{18}w^{3}+14978550732649801624600020yz^{17}w^{4}+13158897289776853139704410yz^{16}w^{5}-28600413702963054911098074yz^{15}w^{6}-3773429365008002179120410yz^{14}w^{7}+26066946048818328242787000yz^{13}w^{8}-11009041595922988846832670yz^{12}w^{9}-4143795761378680174040988yz^{11}w^{10}+4195830877816626077711781yz^{10}w^{11}-1059472573422586970968750yz^{9}w^{12}+72638540163491706593750yz^{8}w^{13}+4227470333314736062500yz^{7}w^{14}+954142267065891921875yz^{6}w^{15}-552489542008989062500yz^{5}w^{16}+96436195541480609375yz^{4}w^{17}-8141225795573406250yz^{3}w^{18}+179035115843375000yz^{2}w^{19}+20928532681031250yzw^{20}-1215089085390625yw^{21}-3722464110877270349669100z^{22}-837403955644663990578600z^{21}w+48845100365033831893680822z^{20}w^{2}-74739214108472119866816750z^{19}w^{3}-49224032526642297903272430z^{18}w^{4}+182067655001132167471880874z^{17}w^{5}-85135362698058877016558361z^{16}w^{6}-93499987116132042815147262z^{15}w^{7}+102529432660573493300575740z^{14}w^{8}-13869257813320947542319870z^{13}w^{9}-22072513275342753255865734z^{12}w^{10}+11200485566465106323091636z^{11}w^{11}-1237311793506100251960361z^{10}w^{12}-445759094924727184218750z^{9}w^{13}+164913328755154939265625z^{8}w^{14}-25350438106480966187500z^{7}w^{15}+2655548029978932687500z^{6}w^{16}-204971194397711500000z^{5}w^{17}+6408057205174625000z^{4}w^{18}+555531758465437500z^{3}w^{19}-71134043164828125z^{2}w^{20}+2430178170781250zw^{21})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 60.96.1.f.2 :

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

Equation of the image curve:

$0$ $=$ $ 11X^{4}+X^{3}Y-X^{2}Y^{2}-22X^{3}Z-3X^{2}YZ+4XY^{2}Z+24X^{2}Z^{2}-3XYZ^{2}-Y^{2}Z^{2}-22XZ^{3}+YZ^{3}+11Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
12.96.0-12.a.2.9 $12$ $2$ $2$ $0$ $0$ full Jacobian
60.96.0-12.a.2.13 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.96.0-60.a.1.18 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.96.0-60.a.1.31 $60$ $2$ $2$ $0$ $0$ full Jacobian
60.96.1-60.b.1.6 $60$ $2$ $2$ $1$ $0$ dimension zero
60.96.1-60.b.1.10 $60$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.384.5-60.g.1.3 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.384.5-60.h.3.8 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.m.4.1 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.o.4.8 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.384.5-60.q.2.6 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.384.5-60.r.2.3 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.w.1.7 $60$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
60.384.5-60.y.1.4 $60$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
60.576.9-60.d.1.15 $60$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
60.960.33-60.j.2.11 $60$ $5$ $5$ $33$ $1$ $1^{16}\cdot8^{2}$
60.1152.33-60.j.1.28 $60$ $6$ $6$ $33$ $1$ $1^{16}\cdot8^{2}$
60.1920.65-60.j.3.21 $60$ $10$ $10$ $65$ $1$ $1^{32}\cdot8^{4}$
120.384.5-120.iu.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ja.3.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ke.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.ks.3.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.og.3.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.om.2.7 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.pq.3.14 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.qe.2.7 $120$ $2$ $2$ $5$ $?$ not computed