curve #6534

y² + xy = x³ − 2683271419743643911571415248437680644411114074492848524217257830790710x + 55749809625327992708980272827639998709495346060890650113804084739761273223461464537986168161839288075172

a-invariants: [1, 0, 0, -2683271419743643911571415248437680644411114074492848524217257830790710, 55749809625327992708980272827639998709495346060890650113804084739761273223461464537986168161839288075172]
c-invariants: [128797028147694907755427931925008670931733475575656729162428375877954081, -48167835516283385700558955723080959078199521218151883331468627102666746462670918724305143035471708713879729]
rank (lower bound): ≥ 32
torsion subgroup: trivial
conductor (N): 540732954631460186687519879844070977651797469333221010708944911078224018220070645337757485138446919694613461294489873333841125454884158962587899148503890045324021788628019990 ★ record for rank ≥ 32
discriminant (Δ): -106231717018299437165540445944589634654468519517374082673183824195333503851556559641527809852556967552893842940439874167763176399649985275957695578435790170715438620129045481036624597364820400907340963840000000 ★ record for rank ≥ 32
Faltings height: 38.8696 ★ record for rank ≥ 32
naive height: 491.2923 ★ record for rank ≥ 32
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 31, 79, 149, 157, 380402823035788592409424989009502439063, 10904186238891380351757305403053529415997265446238367, 17617584604638292489874806912848713825027861369286440683591469060089
regulator: 649610119689629564788012717765700970268521.837225215473585118465623298754430870006503456275885365764410
submitted by: Lewis Combes
submitted at: 2026-11-04 05:59:04 UTC
last updated: 2026-11-04 05:59:04 UTC

Witness: 32 independent points log in to add more points →

(-36509688622188494800034055650857362, -10249358778413674756569849981231538541324544711633324)
(43493303271843225945930082231122834, -4617395067168173573725521461897954897757040583379042)
(17521245565803419906996015763420684, -3756922973738973981664687547664638138272068126404742)
(-227290177274530297399091456690228049/4, -39800623133508125816489324759878573462275093738594771/8)
(27182803956453679448585541603206484, 1701903000251152434772913234854302205516734945848658)
(9875965533581339137056223968811524, 5496650148501520255961227828760086213355538035344098)
(57567303693885358535593094763703284, 9594729496901585517107839040638274043341190170795058)
(-43166271072938544515041939716012036, -9546923824506415798701246324283422160761022579799862)
(47892827449266836621605218422243084, -6090421216628652392112177759898072484951056428689542)
(154203696337269031472100996919460625516/49, -1914624992289434485191051023536275204368794707255513284106/343)
(34429208855917635927910740421834524, -2044080072752756635847123795672363218631484248081622)
(43161067139932712565454543925439494, -4510071166318883161421141404779816074870875144323682)
(3186484026988584742061538341755561644/25, -5280317737465265674794612824355930122622481195037379438/125)
(31219618944209703960052505009989115244/841, -65882563047296873472994589578519369723271538431912529438/24389)
(48358748449107423621271377425095884, 6251432662465873325824823066511340768731682814619258)
(97569956632258992534787503445234014, 26884921723739513215842539038554853519706462271693438)
(-53136179809639901015415245833246836, 6949896678085513850991029053618839262734395531637498)
(267613437194937498535873838450403084, -136026941985508846725890007438000619563405048536209542)
(66278467767637107113648057618019084, 13002195332789902688125634300309606935003918343278458)
(42928442250160129068683063604035514, -4435298950755773236708414204966564313111242232162562)
(7773012401844259620390352646103244524/361, -19347448501136773456456488060558831788478887116254131778/6859)
(-14792194618030993241767270714415988, 9602323523357576404533090406057904113933612354701690)
(-30578091374051302223794216404441086406/1681, 684162301575231678835101102315365457406285929116710878888/68921)
(-409164173435626799361292048775395441076385241374/13354515110689, 509981125473713230418242108407792332050298259120420518801737102252634496/48802512993744999887)
(-36949912069896009584336375852877253256724/1885129, -25987888494074639541162335337784609942336601277028245117057054/2588282117)
(6469512324166004116264271133889469264594653344124716/267797859320723449, 310685662601366034497553500664786676938284039045697426682326527758098134448306/138583221161228871752498957)
(9188385812725596492398695525198522476/169, 18457160862387450771852974706843878352391852899238985346/2197)
(1056004272250067907675941338058224269376330326636/118637472078889, -7374764463399759864970847647361028718741535194930320526377129687785337474/1292209192793441535787)
(83053665659905495617716619204988044, -20144257437883974114780793559708912255911436025509382)
(23851676913958866380302923938324838629307756/867479209, -42436121353997060410918882101321966741271840725853480536878030334/25549865142677)
(44808482519648774575486008185808967149076/2331729, -11959594439194734605770413575863270164590224200347273494459906/3560550183)
(-6226091171857714015396859947042457632053805955104/113709101921809, 7524889112161585849661515310747356159582565092425905799339315458868976874/1212530981760808415623)

Commentary

E has trivial torsion and rank at least 32. It is the fibre at t = -248188/1981293 of a rank-17 elliptic K3 surface (the same surface as #4239). Assuming GRH, a Selmer-group bound computed with conditional class groups gives rank at most 32, so the rank is exactly 32 under GRH. Found by a collaboration between: Blair Butler, Lewis Combes, Shir Levav-Porat Search performed by Claude code Opus 5.5

last edited by Lewis Combes at 2026-11-04 05:59:04 UTC · history