curve #614

y² + xy + y = x³ − x² − 11452843610782327610954979208496829603764690244072x + 14243815623171156762050763778903415193126318778717824544354932805745664171

a-invariants: [1, -1, 1, -11452843610782327610954979208496829603764690244072, 14243815623171156762050763778903415193126318778717824544354932805745664171]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 2798269498065207394410944473876126988677958069177633823142936526529755081283469905582705125459068762952810
discriminant (Δ): 8496633763681602986126483684328449209606559915363493817242256656881705706351913688069631749153061117732813058370185702482394272418885502560000000000
Faltings height: 27.1430
naive height: 350.5006
primes of bad reduction: 2, 5, 7, 11, 13, 17, 23, 31, 89, 149, 1739164791642025923260593499972456814701256936287713873632410489930947782544231790391332891701
regulator: 747547623385888471286509938448.52401385146123194226000538219886548688450422453829987861014549
submitted by: wgxli
submitted at: 2026-09-06 17:31:55 UTC
last updated: 2026-09-06 17:31:55 UTC

Witness: 27 independent points log in to add more points →

(-4919531481030983509731650737785401499/6791606057041, 83318304213047412993116057511050584039203720496400834759/17699407588678895911)
(-92604653232710617402453895901155715539/31493634862561, 838291367738780095645376382543349430348972007414058760631/176739727864268464559)
(427837905478379164672811, 3069555131826942851114820139698529219)
(-3075868309755963356652439, -4513376704518738492276347143357924031)
(-3262716283631332840503459, -4108350527812058487699831178356622951)
(4702856026440551626664587841, -322509426313076141612503097908480299675151)
(-194895377574442837425986296000081398339/57236018571601, -1606271049157921796940448345752640006009862260584074282319/433016179466500213849)
(75614125135705723447468376278762578169/30910008344329, 159931561544298183800388980147667465816436169469313663377/171849662461774021733)
(2185226457403182176958624848699363384989/1422024717126049, 28021548518955388706048347871585165302644781522528209553383/53624186622471006395407)
(2538426007466915776146134464685580329334362600269/5846002883815664342529, 4044209174276859814406381687392099690166944696003176713475945037801686073/446980459758076310122067296092417)
(1742445392443663488854166621666365841581/1215443481230521, 37217755313786720001628056992444034560194416223066264986539/42374262544714193262931)
(18533491483175293144679025401/11881, 540353290923233812423149274604833870354241/1295029)
(3512247847804277603311561, 4164765977422707324888699723059420469)
(93954453247575801316944869975815368550459/106103998374071569, 75695352485945980320966908796629764019517596661918970794708407/34561901245424435144010503)
(-10017269798919932239249906504458022725791/4646615416891569, 1703197630799078826037129378027000727585899851650571669070907/316741590763371962020503)
(1306298613840199878523878227609/299209, -1127498977329044634747905246609428424153707793/163667323)
(72250517990150990080611328723539162709/1814912812969, -611978068230931515071289767841849086599128462451324251693/2445026947765268203)
(1698610059579846992032988487353689084407975781/573741476423925266521, -34456848737744023426185292256183920671234121627151536214846394487861/13742772905435567465824358656931)
(75786151599963029721119425769021361/24266162176, -11300973732198175608498726069220492181320871905798231/3780085679128576)
(29068378504350887528681790761261212363879/2224927957016164, -4803204651855417327257132995855495683893982347129622240822803/104947942744265336011288)
(1582122508746976738090420257245447334381/21793622436496, -62863109045592971602252225450867694069539587758895563283489/101740562412130212544)
(10763500008156328169977511, 33733576336232390911577769965937762169)
(876470363871354966453624734258449/193293409, 20028130845514130528771323955150040630442827501363/2687358265327)
(2327770610482977268130414620465/994009, -511165456088127661029287981046629321331902019/991026973)
(179059309790906605061357351134178148033151/19083839610121441, 71366483003953235287508220869387029914547096443579582995433159/2636323110821150688414511)
(2596519570692466674539662105879323569726699/782605794122109409, 2473764960423786066714791774067375107579393891246856296517058387/692332291682886118516462673)
(291582574245301598136331486362901709/74811549289, 109837464098239009861126842533775368569126790138020477/20462230526879413)

Commentary

Specialization at T = 7498/65 of the rank-15 fibration of X948 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 76564T^2 + 12654410T + 386426887, p = 39(T - 277)(8T - 207), q = 988(17455T + 1023958), D = 65(8T - 207)(1431540T^3 + 302133923T^2 + 13245112857T + 4685553964), E = -416(741334690T^3 + 43193232633T^2 - 414544705249T - 21459282173892).

last edited by wgxli at 2026-09-17 17:26:15 UTC · history