Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x y + y^{2} - z^{2} $ |
| $=$ | $5 x y + 15 x z + 6 y^{2} + 2 y w + 10 z^{2} + w^{2}$ |
| $=$ | $15 x^{2} + 9 x y - 30 x z + 11 y^{2} + 4 y w + 21 z^{2} + 2 w^{2} - t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 109 x^{8} - 37 x^{7} y + 41 x^{6} y^{2} - 3615 x^{6} z^{2} - 8 x^{5} y^{3} + 660 x^{5} y z^{2} + \cdots + 202500 z^{8} $ |
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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 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^{10}\cdot3^3\cdot5^3\cdot109^2}\cdot\frac{5009165341949020849069654366526402716845058593750xzw^{16}-5063182678325125660444093182688145591518117968750xzw^{14}t^{2}+1887497078936620714546439541778302550283161875000xzw^{12}t^{4}-101936745662208420509997338422092685339404000000xzw^{10}t^{6}-438256170824819407136773769398078373639670000000xzw^{8}t^{8}+144577297963603928597086448305674436549612800000xzw^{6}t^{10}+60361752089270156389186351237223213866762944000xzw^{4}t^{12}-20171576240736515762404318821643719038401324800xzw^{2}t^{14}-231980521613681819343649718887610689068860160xzt^{16}+159792588561716695962495657726131529325775390625xw^{17}-160577023867350655841017996045380563349383203125xw^{15}t^{2}+60017378674130165901865475363855931370219218750xw^{13}t^{4}-6358833811741258753168878851402334066581250000xw^{11}t^{6}-7597234177035432230812689960490099208554875000xw^{9}t^{8}-476159658424041030969340572216424587793980000xw^{7}t^{10}+3696969445767710010391381318784713416057576000xw^{5}t^{12}-889003652441102231485750006562163978828854400xw^{3}t^{14}-1620220280351518079258903964629322735303210937500yzw^{16}+1635248351822177033890542449653064699724089062500yzw^{14}t^{2}-614485131222296788701192890862679981096126875000yzw^{12}t^{4}+61042673206430293185315941052100836883207500000yzw^{10}t^{6}+83262120527506695237084876466463672971747500000yzw^{8}t^{8}+3910152890337370807608165070897630740958320000yzw^{6}t^{10}-38456869733412477674490769254552040018535136000yzw^{4}t^{12}+9285558473794292999009207131714995240840729600yzw^{2}t^{14}-10721030640539706572273499210814381061798400yzt^{16}+661464462749443539315114129460705567348558593750yw^{17}-679555277227659011185280980001527021337544140625yw^{15}t^{2}+260825662344249730936885912886741872529111484375yw^{13}t^{4}-20193637663781693396635120435360678687673906250yw^{11}t^{6}-53335819596294340185890732999543835638274000000yw^{9}t^{8}+16533558667148540440642474650122867180210865000yw^{7}t^{10}+9223579229196059009533249443832934663542980000yw^{5}t^{12}-3168203928029945709666564590924407287552388800yw^{3}t^{14}+51378311158696757851998873017974035977078400ywt^{16}-12961762242812144634071231717034581882425687500000z^{3}w^{15}+12963074466174526331961991648486397949550275000000z^{3}w^{13}t^{2}-4812861613714840695710917977175858051280970000000z^{3}w^{11}t^{4}+452563169248863577519160003970459524096040000000z^{3}w^{9}t^{6}+683245166879448327199083776986130581812840000000z^{3}w^{7}t^{8}+10296103960023134807476627877402927628721920000z^{3}w^{5}t^{10}-295659038996360109570112825329636749838263808000z^{3}w^{3}t^{12}+71953215076617924003872183876210035980906086400z^{3}wt^{14}+5366519893987797440029647536295822572410476562500z^{2}w^{16}-5432000792390509697795163523109680845592101562500z^{2}w^{14}t^{2}+2027448858241780685042724759244292216926239375000z^{2}w^{12}t^{4}-131631489010816929641970110361963662817907500000z^{2}w^{10}t^{6}-429650499061281488976691027458614227834231500000z^{2}w^{8}t^{8}+126664849632315014681429840277483560196530640000z^{2}w^{6}t^{10}+73771993848399109947559226686319687743083360000z^{2}w^{4}t^{12}-22973002414208451011194460187396210678596889600z^{2}w^{2}t^{14}-166968817228761505000288329972047248767398400z^{2}t^{16}-810110140175759039629451982314661367651605468750zw^{17}+973021934264132503870242738295942555241459765625zw^{15}t^{2}-463849363984365620106064600102109264813794453125zw^{13}t^{4}+88669849142112766379361258238114731969652968750zw^{11}t^{6}+36991713628510674558412185591988884630873750000zw^{9}t^{8}-7638884183218622271643342965489840935975115000zw^{7}t^{10}-18737875884388239037502405425385878762965276000zw^{5}t^{12}+8176237333526316859767803866644410878917345600zw^{3}t^{14}-889003652441102231485750006562163978828854400zwt^{16}+334448187906118605030607973077524078681855468750w^{18}-338037270466678110309274567738078643196855468750w^{16}t^{2}+126157760372206069124327854909830902487043125000w^{14}t^{4}-8230792414690783156880951439699122016063750000w^{12}t^{6}-26509421559342733250692139487529021869827400000w^{10}t^{8}+7344934671096306435634747630844681178740880000w^{8}t^{10}+5010793942535574994555492131173999609242560000w^{6}t^{12}-1565373261201509626625503789564119348031948800w^{4}t^{14}+8564342802800532952163178984548233630790400w^{2}t^{16}-1001237954599138371183603899043983870918656t^{18}}{t^{2}(179076552711218799414437137500000xzw^{14}-11044447195305503662202374200000xzw^{12}t^{2}+36297645707311003922672810916000xzw^{10}t^{4}+105882587950694165723984774256000xzw^{8}t^{6}-129284198448255414187214062022550xzw^{6}t^{8}-931513305094465375304864967300xzw^{4}t^{10}+241413617859935859479511250530xzw^{2}t^{12}+44342458306449948568059240xzt^{14}+6781030790597398833027176250000xw^{15}+270518169207709486473688800000xw^{13}t^{2}+1703968134347448809443552050000xw^{11}t^{4}+3657450005901619609561308072000xw^{9}t^{6}-4021924713554143955171801053665xw^{7}t^{8}-30602945270061655308272192654xw^{5}t^{10}+7445490831746447283884997765xw^{3}t^{12}-65688181029020998887092550000000yzw^{14}+40302507602472654466591200000yzw^{12}t^{2}-15582835654643638164019549680000yzw^{10}t^{4}-36195947108349276936583815840000yzw^{8}t^{6}+41146531738495567518981614785500yzw^{6}t^{8}+168730482655578468402479690540yzw^{4}t^{10}-74217303121687147410758812840yzw^{2}t^{12}+38922116799569612142210960yzt^{14}+24772101150419944233511275000000yw^{15}-1526543774577351390105865350000yw^{13}t^{2}+5036758295789482963268473440000yw^{11}t^{4}+14073393465725935021944565626000yw^{9}t^{6}-17207071810777558999838468394870yw^{7}t^{8}+170245465327803283892171393881yw^{5}t^{10}+33244091211217738014718814730yw^{3}t^{12}-527104445946649634377898319ywt^{14}-525505448232167991096740400000000z^{3}w^{13}-2186403418614513269437862400000z^{3}w^{11}t^{2}-128017463462094516870345125760000z^{3}w^{9}t^{4}-292982682698183480210357308416000z^{3}w^{7}t^{6}+327270583460794566112508901592800z^{3}w^{5}t^{8}+2254985951101564861011582844800z^{3}w^{3}t^{10}-601484543377730312528166505920z^{3}wt^{12}+198176809203359553868090200000000z^{2}w^{14}-7833138128017411195040752800000z^{2}w^{12}t^{2}+40327394337614433039677835840000z^{2}w^{10}t^{4}+113648803218386595969504407136000z^{2}w^{8}t^{6}-137355868772645036167129456674180z^{2}w^{6}t^{8}-1066658596007358564645833879380z^{2}w^{4}t^{10}+260037671576445179709387259440z^{2}w^{2}t^{12}+71841311169138197680839120z^{2}t^{14}-32844090514510499443546275000000zw^{15}+6305297904337738944382745850000zw^{13}t^{2}-8009254193162089529311526160000zw^{11}t^{4}-16736876967552157490129469342000zw^{9}t^{6}+24071839006095856541520403296750zw^{7}t^{8}-3867640613815333768735041453953zw^{5}t^{10}-68363416202514734688550821030zw^{3}t^{12}+7445490831746447283884997765zwt^{14}+12386050575209972116755637500000w^{16}-601474564949934571971670800000w^{14}t^{2}+2451411188750302588389473820000w^{12}t^{4}+7071453057576109823507058465600w^{10}t^{6}-8615470012998414113795286461070w^{8}t^{8}-62358163209857224056633696600w^{6}t^{10}+16092407933804878082026564710w^{4}t^{12}+2956163887096663237870616w^{2}t^{14})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.5.qe.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{15}t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 109X^{8}-37X^{7}Y+41X^{6}Y^{2}-3615X^{6}Z^{2}-8X^{5}Y^{3}+660X^{5}YZ^{2}+4X^{4}Y^{4}-660X^{4}Y^{2}Z^{2}+39150X^{4}Z^{4}-1575X^{3}YZ^{4}+1575X^{2}Y^{2}Z^{4}-155250X^{2}Z^{6}+202500Z^{8} $ |
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.
