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Properties

Label 60.80.5.o.1
Level $60$
Index $80$
Genus $5$
Analytic rank $4$
Cusps $4$
$\Q$-cusps $0$

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Invariants

Level: $60$ $\SL_2$-level: $20$ Newform level: $3600$
Index: $80$ $\PSL_2$-index:$80$
Genus: $5 = 1 + \frac{ 80 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (none of which are rational) Cusp widths $20^{4}$ Cusp orbits $4$
Elliptic points: $0$ of order $2$ and $2$ of order $3$
Analytic rank: $4$
$\Q$-gonality: $4$
$\overline{\Q}$-gonality: $4$
Rational cusps: $0$
Rational CM points: yes $\quad(D =$ $-3$)

Other labels

Cummins and Pauli (CP) label: 20B5
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 60.80.5.6

Level structure

$\GL_2(\Z/60\Z)$-generators: $\begin{bmatrix}8&47\\37&59\end{bmatrix}$, $\begin{bmatrix}26&3\\53&5\end{bmatrix}$, $\begin{bmatrix}42&11\\5&13\end{bmatrix}$, $\begin{bmatrix}58&11\\13&42\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: none in database
Cyclic 60-isogeny field degree: $144$
Cyclic 60-torsion field degree: $2304$
Full 60-torsion field degree: $27648$

Jacobian

Conductor: $2^{20}\cdot3^{6}\cdot5^{10}$
Simple: no
Squarefree: yes
Decomposition: $1^{5}$
Newforms: 400.2.a.a, 400.2.a.e, 3600.2.a.bb, 3600.2.a.bf, 3600.2.a.k

Models

Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations

$ 0 $ $=$ $ 5 x^{2} - x w - x t + y^{2} - 4 y z - 2 y w + z^{2} - 2 z w - 3 w^{2} - 2 w t - t^{2} $
$=$ $2 x^{2} + x y + 2 x z - 3 x w + 4 x t + y^{2} - 3 y w - y t - z^{2} + 3 z w + z t$
$=$ $3 x^{2} + 4 x y - 3 x z - 4 x w + x t - y^{2} + 3 y z - 6 y w - y t - 2 z^{2} - 3 z w - 5 w t - t^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 78070476 x^{8} - 590200680 x^{7} y + 378728900 x^{7} z + 1670261340 x^{6} y^{2} - 2818465740 x^{6} y z + \cdots - 21879 z^{8} $
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Rational points

This modular curve has 1 rational CM point but no rational cusps or other known rational points.

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x+\frac{6}{11}w+2t$
$\displaystyle Y$ $=$ $\displaystyle t$
$\displaystyle Z$ $=$ $\displaystyle y+z-\frac{30}{11}w$

Maps to other modular curves

$j$-invariant map of degree 80 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 3\cdot5^2\,\frac{2233050297185715925400042030xzw^{9}+7351630424170763636300321539xzw^{8}t-1567842171282489868117604510xzw^{7}t^{2}-25134124364138105417125537812xzw^{6}t^{3}-34152454686027085932042029670xzw^{5}t^{4}-17384765309132969078274310290xzw^{4}t^{5}+2450594398580950334223776022xzw^{3}t^{6}+7756787260498232260083610404xzw^{2}t^{7}+3945737539657369879359102000xzwt^{8}+757434067939682188989891615xzt^{9}+3800754296859745319918272370xw^{10}-10078648489275736193405843567xw^{9}t-5786904826279366952908285730xw^{8}t^{2}+22515658053478737484863207236xw^{7}t^{3}+17044080841520909377644192750xw^{6}t^{4}-20961465919170883858498239750xw^{5}t^{5}-32705504456971171341234682206xw^{4}t^{6}-11091540636313419815302779252xw^{3}t^{7}+6131465792918068913452316520xw^{2}t^{8}+6058807665152129844645582405xwt^{9}+1570969703211487699583680200xt^{10}+3384258836787462104166832270yzw^{9}+13816809493524532626176677553yzw^{8}t-829121485736385967072904450yzw^{7}t^{2}-44921108674664564454663963324yzw^{6}t^{3}-63563231297915400713360022690yzw^{5}t^{4}-33154904424060279472963429350yzw^{4}t^{5}+4192036837899625234883488554yzw^{3}t^{6}+14454913593385982179929571308yzw^{2}t^{7}+7389282624688056720285791100yzwt^{8}+1412512357930310260400317605yzt^{9}-1506192037073446123504908100yw^{10}-753334895343914772020865794yw^{9}t+2141452101382263673140478093yw^{8}t^{2}+5081149429601875324269955662yw^{7}t^{3}+7740222549736963727067063036yw^{6}t^{4}+7917145566743214450045389790yw^{5}t^{5}+3990442555398065099857881858yw^{4}t^{6}-576380617668258833023476390yw^{3}t^{7}-2047338554368363242390475272yw^{2}t^{8}-1150275514260355130480665140ywt^{9}-250347927445285487076457695yt^{10}+982507443229612346046920040z^{3}w^{8}+678600764795667063704412300z^{3}w^{7}t-2469225060301127578829733420z^{3}w^{6}t^{2}-4632845521247091938629055700z^{3}w^{5}t^{3}-2917358575375741396838541900z^{3}w^{4}t^{4}+13246311432637558452630420z^{3}w^{3}t^{5}+1070885243472052167417085740z^{3}w^{2}t^{6}+612356745532384212016288500z^{3}wt^{7}+126113689178834295505356900z^{3}t^{8}+4811414370193287452878189570z^{2}w^{9}-1804814266565727370599335443z^{2}w^{8}t-15530844697284947472814660070z^{2}w^{7}t^{2}-9779247220576184108823436656z^{2}w^{6}t^{3}+9575247474850084818552521490z^{2}w^{5}t^{4}+14865016531683269677336139070z^{2}w^{4}t^{5}+5015704635475475976053940366z^{2}w^{3}t^{6}-2505590240235080191864681248z^{2}w^{2}t^{7}-2481491692216204462298990700z^{2}wt^{8}-628970899636565921307637755z^{2}t^{9}-11737718884178732308555910650zw^{10}+5297500197665996873268382885zw^{9}t+42078183685375909965722352006zw^{8}t^{2}+23961032124531858336012550040zw^{7}t^{3}-36660800402879127016166697198zw^{6}t^{4}-55568703134772854665294711530zw^{5}t^{5}-22451959745530909334101402290zw^{4}t^{6}+7331443289292573364558420728zw^{3}t^{7}+10455242314454654765882189616zw^{2}t^{8}+3785489025724482252221085525zwt^{9}+425516282995039060236281460zt^{10}-3674360402491152177422981430w^{11}-2569917817816322293191879835w^{10}t+11192941031043592957834104385w^{9}t^{2}+21344721911976831919814770364w^{8}t^{3}+7838101013708118811681726870w^{7}t^{4}-15721759391041341152515465812w^{6}t^{5}-21834475603345342788960451380w^{5}t^{6}-9724169972735908970210140230w^{4}t^{7}+1780261448055770123557081272w^{3}t^{8}+3989959344800570441776338999w^{2}t^{9}+1805905280078614466896675275wt^{10}+313027904405725994225172690t^{11}}{302039729837586030xzw^{9}+3154104024696367177xzw^{8}t+2980604476321848210xzw^{7}t^{2}-2071104331647836976xzw^{6}t^{3}-4103046331731110754xzw^{5}t^{4}-2186079490575354330xzw^{4}t^{5}-456510087644349330xzw^{3}t^{6}+16608321265940928xzw^{2}t^{7}+22434236116118532xzwt^{8}+3116354031717777xzt^{9}-507940954312467210xw^{10}-3766451671105785281xw^{9}t+4301762864965829390xw^{8}t^{2}+11823463097733036168xw^{7}t^{3}+2597755182687883482xw^{6}t^{4}-5638649381754741630xw^{5}t^{5}-4124437396833374190xw^{4}t^{6}-991089303953681304xw^{3}t^{7}+30439148509432224xw^{2}t^{8}+51634761230831679xwt^{9}+7623157510740960xt^{10}+360602144048886890yzw^{9}+1241501458939545379yzw^{8}t-3496202437559329170yzw^{7}t^{2}-9952364769674265312yzw^{6}t^{3}-8406258429016796958yzw^{5}t^{4}-2900133823243512510yzw^{4}t^{5}-100760139071262990yzw^{3}t^{6}+239543451712284336yzw^{2}t^{7}+71396411512115844yzwt^{8}+7584559743925179yzt^{9}-682398616563585440yw^{10}-2054854511827561762yw^{9}t-1852680721725647241yw^{8}t^{2}+1552221844027625526yw^{7}t^{3}+4539651526603823112yw^{6}t^{4}+3851314069723367202yw^{5}t^{5}+1593669142327632930yw^{4}t^{6}+306272507202907722yw^{3}t^{7}-541729032501816yw^{2}t^{8}-10287637077717648ywt^{9}-1430067806932761yt^{10}+18324643337699820z^{3}w^{8}-287259724919564100z^{3}w^{7}t-642990886569696060z^{3}w^{6}t^{2}-493095565492126740z^{3}w^{5}t^{3}-139291863805383300z^{3}w^{4}t^{4}+16466918209780500z^{3}w^{3}t^{5}+21929687690052780z^{3}w^{2}t^{6}+5639370504449220z^{3}wt^{7}+568252124934120z^{3}t^{8}-132613771199945470z^{2}w^{9}-3085405558378543949z^{2}w^{8}t-3495246072667730190z^{2}w^{7}t^{2}+1052918211687135132z^{2}w^{6}t^{3}+3213005298507392598z^{2}w^{5}t^{4}+1712502133050418110z^{2}w^{4}t^{5}+298301952487748670z^{2}w^{3}t^{6}-51488717348451996z^{2}w^{2}t^{7}-27012026685817944z^{2}wt^{8}-3450151702043649z^{2}t^{9}+231107190011462650zw^{10}+9618340472356919615zw^{9}t+14341744909162804058zw^{8}t^{2}+1786162018689312780zw^{7}t^{3}-8649880063049690994zw^{6}t^{4}-6873941515378108266zw^{5}t^{5}-2153434491830990970zw^{4}t^{6}-157149468313744140zw^{3}t^{7}+80301605045956632zw^{2}t^{8}+22987157451160923zwt^{9}+1939210829410608zt^{10}+69512325864715210w^{11}+2463408660621540495w^{10}t+6051815413520865395w^{9}t^{2}+5886146682635018932w^{8}t^{3}+1413196949176779510w^{7}t^{4}-2301011169939354696w^{6}t^{5}-2351788122358804284w^{5}t^{6}-907182850313014290w^{4}t^{7}-111905734304520720w^{3}t^{8}+30807064924188633w^{2}t^{9}+12179632820002137wt^{10}+1371979297288062t^{11}}$

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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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{\mathrm{ns}}^+(4)$ $4$ $20$ $20$ $0$ $0$ full Jacobian
15.20.0.b.1 $15$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
15.20.0.b.1 $15$ $4$ $4$ $0$ $0$ full Jacobian
20.40.2.e.1 $20$ $2$ $2$ $2$ $2$ $1^{3}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
60.160.9.q.1 $60$ $2$ $2$ $9$ $5$ $1^{4}$
60.160.9.s.1 $60$ $2$ $2$ $9$ $4$ $1^{4}$
60.160.9.bm.1 $60$ $2$ $2$ $9$ $7$ $1^{4}$
60.160.9.bn.1 $60$ $2$ $2$ $9$ $6$ $1^{4}$
60.160.9.dh.1 $60$ $2$ $2$ $9$ $5$ $1^{4}$
60.160.9.di.1 $60$ $2$ $2$ $9$ $4$ $1^{4}$
60.160.9.dp.1 $60$ $2$ $2$ $9$ $5$ $1^{4}$
60.160.9.dq.1 $60$ $2$ $2$ $9$ $4$ $1^{4}$
60.240.15.gw.1 $60$ $3$ $3$ $15$ $9$ $1^{10}$
60.240.15.jo.1 $60$ $3$ $3$ $15$ $9$ $1^{10}$
60.240.19.di.1 $60$ $3$ $3$ $19$ $12$ $1^{12}\cdot2$
60.320.23.o.1 $60$ $4$ $4$ $23$ $11$ $1^{18}$
120.160.9.cj.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.cp.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.fg.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.fj.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.ls.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.lv.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.mq.1 $120$ $2$ $2$ $9$ $?$ not computed
120.160.9.mt.1 $120$ $2$ $2$ $9$ $?$ not computed
120.320.23.bk.1 $120$ $4$ $4$ $23$ $?$ not computed