$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}1&30\\0&13\end{bmatrix}$, $\begin{bmatrix}7&15\\52&41\end{bmatrix}$, $\begin{bmatrix}17&50\\20&57\end{bmatrix}$, $\begin{bmatrix}21&50\\52&19\end{bmatrix}$, $\begin{bmatrix}57&55\\16&47\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.144.1-60.m.1.1, 60.144.1-60.m.1.2, 60.144.1-60.m.1.3, 60.144.1-60.m.1.4, 60.144.1-60.m.1.5, 60.144.1-60.m.1.6, 60.144.1-60.m.1.7, 60.144.1-60.m.1.8, 120.144.1-60.m.1.1, 120.144.1-60.m.1.2, 120.144.1-60.m.1.3, 120.144.1-60.m.1.4, 120.144.1-60.m.1.5, 120.144.1-60.m.1.6, 120.144.1-60.m.1.7, 120.144.1-60.m.1.8, 120.144.1-60.m.1.9, 120.144.1-60.m.1.10, 120.144.1-60.m.1.11, 120.144.1-60.m.1.12, 120.144.1-60.m.1.13, 120.144.1-60.m.1.14, 120.144.1-60.m.1.15, 120.144.1-60.m.1.16, 120.144.1-60.m.1.17, 120.144.1-60.m.1.18, 120.144.1-60.m.1.19, 120.144.1-60.m.1.20, 120.144.1-60.m.1.21, 120.144.1-60.m.1.22, 120.144.1-60.m.1.23, 120.144.1-60.m.1.24 |
Cyclic 60-isogeny field degree: |
$4$ |
Cyclic 60-torsion field degree: |
$64$ |
Full 60-torsion field degree: |
$30720$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} + x y - 2 x z + 3 x w + 2 y w - 4 z w + w^{2} $ |
| $=$ | $2 x^{2} - 12 x y + 9 x z + 2 x w - 10 y^{2} + 10 y z + y w - 10 z^{2} - 2 z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 1504 x^{4} - 270 x^{3} y + 556 x^{3} z + 15 x^{2} y^{2} - 330 x^{2} y z + 204 x^{2} z^{2} + 30 x y^{2} z + \cdots + 4 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
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{3}{2^{21}}\cdot\frac{950832005837866811412080504851632508709085xz^{17}+2983187288993240173725763759126964345477610xz^{16}w-149530810653584819484805500528914168100532728xz^{15}w^{2}+153648239966347854194259013604859908578927200xz^{14}w^{3}+3624646747489939870020172320107058691268762652xz^{13}w^{4}-7241215954156551683112693958941407728283827032xz^{12}w^{5}+4215496166197727608716859612987395604405353432xz^{11}w^{6}+79844347440034578986677934786642613559247616xz^{10}w^{7}-1001022551245857939040587842212804148914588770xz^{9}w^{8}+379307985066653509514089009189461172223627532xz^{8}w^{9}-24752353102745054756038567979133402125881992xz^{7}w^{10}-19659365582520161710007533685743614257063328xz^{6}w^{11}+6461920978024828001272015705117490988025596xz^{5}w^{12}-902601721265739092559815783407356922228920xz^{4}w^{13}+58599685252607389551256959928441576234536xz^{3}w^{14}-575174755875937038963301621800150457536xz^{2}w^{15}-128410891598169864150501649673423525139xzw^{16}+4925392010085934867975599654630383130xw^{17}+209261674861168029619223008700368105774614y^{2}z^{16}-18190507362935355232635957200016355762877088y^{2}z^{15}w+237860368749948642280501717139555442249814368y^{2}z^{14}w^{2}-1721958606504471084120074313478127673572146944y^{2}z^{13}w^{3}+3163538246100834974642370906178720360090421592y^{2}z^{12}w^{4}+620738531933458107213393883735108027461998880y^{2}z^{11}w^{5}-5555210255906699516339072937795623323438365888y^{2}z^{10}w^{6}+5354495135668435264451355844454888023930974528y^{2}z^{9}w^{7}-2408972026338696979072744534255604813672149644y^{2}z^{8}w^{8}+537597343889775913844228202987219226208421792y^{2}z^{7}w^{9}-26582248297826932961170038719400879953933856y^{2}z^{6}w^{10}-16888468445483825346902488581660664597635456y^{2}z^{5}w^{11}+4852354762844832527365814511261400540417464y^{2}z^{4}w^{12}-633226913654838299910407263252574373214752y^{2}z^{3}w^{13}+45055840056028663794818333823911116430976y^{2}z^{2}w^{14}-1642068682621241003586244328068073014464y^{2}zw^{15}+22579308787045929993684550888052785254y^{2}w^{16}-209261674861168029619223008700368105774614yz^{17}+17721705707542634477905955452327310267213673yz^{16}w-188498665772750567409701457388363002899823996yz^{15}w^{2}+1031259409644809448160729046632139551125796884yz^{14}w^{3}-3097134732251663096252727193197543612237679620yz^{13}w^{4}+6385989319888516810911842657390581920337873608yz^{12}w^{5}-6802528377769427426654451299683119041207081580yz^{11}w^{6}+2802948631439833811085623257291726340274432948yz^{10}w^{7}+746709275407566872857464663845104969245109584yz^{9}w^{8}-1343887670745412062178348082177051663474586030yz^{8}w^{9}+665993747950508594530175563869594488271068172yz^{7}w^{10}-181944740976257122246178884158696332815963044yz^{6}w^{11}+29935509663763450761020951311471092453633540yz^{5}w^{12}-2839276307310706144913929772559808800623328yz^{4}w^{13}+112986250569756270809067902089587938428380yz^{3}w^{14}+4235893342916443692248111038182445004892yz^{2}w^{15}-605870336697557102277613337885868014394yzw^{16}+17619097995119755690110057481451569629yw^{17}+209110695606278587065561742661307342528022z^{18}-15926031346591489852874357341325776171902138z^{17}w+200144084646510829105641010108772351096555697z^{16}w^{2}-985212690354739812281980737357269945907846618z^{15}w^{3}+235473112830390417886082516800844389331039814z^{14}w^{4}+1452099541191373097405876577686740563821302638z^{13}w^{5}-206242419065263107453173244076362875824882846z^{12}w^{6}-1455746213966481306543506745117825531570223170z^{11}w^{7}+995732233624210410405920770402846986294351402z^{10}w^{8}-40053473693861159954625723518653598628709246z^{9}w^{9}-217026801663272600963796221710981854122804316z^{8}w^{10}+115739432685938975976363405444962458633793394z^{7}w^{11}-30376725037542886528866824859598044588606302z^{6}w^{12}+4637079534502977534498837054603381452254458z^{5}w^{13}-398342457572573229903930055644443878524498z^{4}w^{14}+13294605886836027310644507026180101267754z^{3}w^{15}+652838614430559162018982984366266836184z^{2}w^{16}-73149237985915373721151835127374729904zw^{17}+1881988844515998025124906433406532251w^{18}}{12751099284536206208409600000000xz^{17}-315447438618496423144404910464000xz^{16}w+430335247016282264680774269135360xz^{15}w^{2}+1030636373041318140211566924637440xz^{14}w^{3}-1906821624669689639746116058038528xz^{13}w^{4}+908139430222874690052975413766336xz^{12}w^{5}+94761060453092958061132198544640xz^{11}w^{6}-241261671278481772380120324014064xz^{10}w^{7}+89818213834651083747550573152624xz^{9}w^{8}-10605803541475677648151532020260xz^{8}w^{9}-2087313377284179053892557586150xz^{7}w^{10}+912481270920005892937687211337xz^{6}w^{11}-127907280311950824901581219396xz^{5}w^{12}+5528203408768061092406001324xz^{4}w^{13}+591881760769272162237158592xz^{3}w^{14}-86186307298827904394841360xz^{2}w^{15}+3744915907592997321500352xzw^{16}-42558757967018053143744xw^{17}-5456743868533536516096000000000y^{2}z^{16}+22459685316742978333400298754048y^{2}z^{15}w+433761386478219655019791070558208y^{2}z^{14}w^{2}-780214898461734352135077437447168y^{2}z^{13}w^{3}+253382815978053273553764162141184y^{2}z^{12}w^{4}+232383203222293252725761766637824y^{2}z^{11}w^{5}-202984151666069929175560250649600y^{2}z^{10}w^{6}+52344768896603251056840923711168y^{2}z^{9}w^{7}+2219404995742369506232885003328y^{2}z^{8}w^{8}-4260304129891690315838489663376y^{2}z^{7}w^{9}+963963169174968012888864573176y^{2}z^{6}w^{10}-72210008170758410452630994332y^{2}z^{5}w^{11}-6085483469860445657466987736y^{2}z^{4}w^{12}+1595912837285102048140859808y^{2}z^{3}w^{13}-112800068694371327644951872y^{2}z^{2}w^{14}+2333218318373007929365312y^{2}zw^{15}+34312187199269582298240y^{2}w^{16}+5456743868533536516096000000000yz^{17}-116806603919069716005438215944192yz^{16}w+107335383572034107488746644292608yz^{15}w^{2}+653631746236079080382471003199488yz^{14}w^{3}-1140777545771936868319121927826176yz^{13}w^{4}+595782255647393052608146933475840yz^{12}w^{5}+17527992717244787799153769266624yz^{11}w^{6}-143294493780899550351677264332352yz^{10}w^{7}+62137883103074952295666654124240yz^{9}w^{8}-9995648427624539240030502164216yz^{8}w^{9}-680850488727392230301422734404yz^{7}w^{10}+572608818018010876679290875884yz^{6}w^{11}-100356993259535626903249025923yz^{5}w^{12}+6946695102549052737934793098yz^{4}w^{13}+145497316924607557735841160yz^{3}w^{14}-54858884877547451772303472yz^{2}w^{15}+3110474523232909890440720yzw^{16}-51136804766835448718304yw^{17}-5456743868533536516096000000000z^{18}+145037174422408896680305415944192z^{17}w+330004068237373241363040474493952z^{16}w^{2}-1944381904051954068920550072987136z^{15}w^{3}+2139830426130516214199263963548672z^{14}w^{4}-507210077482426557272926641936000z^{13}w^{5}-570023860425815578921297686097664z^{12}w^{6}+483482525589396746130637422872288z^{11}w^{7}-145790892246329183305163442360480z^{10}w^{8}+6922597212784171412834673497928z^{9}w^{9}+8407633751800154664501517816372z^{8}w^{10}-2751270490932335152023496334046z^{7}w^{11}+366577806610547714274126296378z^{6}w^{12}-8825024332765364863690926903z^{5}w^{13}-4041941044697956035457440790z^{4}w^{14}+580211320822349030294689864z^{3}w^{15}-32318939093721225166666384z^{2}w^{16}+546765011686010819967184zw^{17}+8578046799817395574560w^{18}}$ 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Hi
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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 $X(1)$.
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.