curve #535

y² + xy = x³ − 9058527942382627845473972898215409636289374648526242485377510x + 10558319813569733256455401415739075542384455012892824206671716720386528743090583309911632100

a-invariants

[1, 0, 0, -9058527942382627845473972898215409636289374648526242485377510, 10558319813569733256455401415739075542384455012892824206671716720386528743090583309911632100]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

689859962667095048106613371898592419800105514953006245767276900156469140962257733608300578016354935743356374554820132232446389774090026266183343010954690

naive height

432.7023

Faltings height

33.9026

discriminant (Δ)

-586387977551608392544174996659292255766160775576093964546366157225477167715924254830548302323256016227034352390036387255036465306905596435615409797931262708366339486301208084480000000

primes of bad reduction

2, 3, 5, 7, 11, 13, 17, 23, 47, 7417, 1626971, 168564758519, 1099998277840213, 558681273133768544784361227429599427271856224404191734197520085496148334778570379209668871970771791864687431

regulator

658695637506123331273903076467612310.9608025298267109073604202554871566236997488589231327539806

submitted by

Mundoamundo

submitted at

2026-09-02 20:40:40 UTC

later contributions

primes of bad reduction recorded · David Renshaw · 2026-09-03 03:25:51 UTC

Witness: 28 independent points log in to add more points →

(892894575116207066937546433120, -1783782253849265034954154991910483165993604310)
(1733106326034023208573997236020, -254136993121217470738352641226388679632180310)
(206929924124764792523679324976780/9, -2952389218968688458001627617981941150782174480370/27)
(1547783951485086322512801736300, 495581774550837106766711377096787392410674890)
(12055848580068776362909926273580/9, -24563929885320481463591666718172076687185804370/27)
(-3426808731153217269050774207380, 1165798255103924547629275229404820882387711690)
(-3477162898489384885961744256448, 122867334621898624320953447778398627719558166)
(9072851240793763176264594212045/16, 151505561984975221716238136456989972354950060785/64)
(-939183195897752540211190563220, -4270540520832053533439910231799630177097165110)
(-30548883036533419710953267336915/16, 292540525352901156725975747762762283994607282785/64)
(1438407187440401644146370565120, -710319999203911222958202985831254429565360810)
(-3173584832658482029181047657330, -2709819286171034464797914805886972077473408860)
(-54682762447132367979232452846643/16, 80886587384367364339113640585840519755169143899/64)
(2046628015743275587548732855020, 769132851104245754581270586183751402610185290)
(1500882648689192732857677322520, 586083430349565642121491017531839939511572190)
(-2737480204309147874200701592270, 3852497555428743516630378540189095970909604640)
(1791445694667160532955135428492, -282321817283909106081051700798106860234261174)
(205688527023354007646709591637018688/134689, -26445080736642401726174394383363126370137862570443062/49430863)
(35517501991172275502283300526753819274744/86025625, 6693482204607611099774114635485409718725318513068192673057978/797887671875)
(-41241283270312700531899767580, -3306332558211171830778735327479898289465311310)
(-284311549607625225815136299128827634708424080/504686603883361, 44612294610880666189454549139137327796593986167962335622574779299190/11337901138845201921391)
(2566278559056681617980638435295100/1681, 36938393590022444354538231065670375713716047379690/68921)
(23897771877055750473562231600258452862551980/5616790980529, 93217478526369628178182413466279006495035631233402447824364279770/13311665437661177833)
(-1462775105293907633355196316943220/529, 46272587850606204517050476585337125882018740784230/12167)
(21890268327551481063838884014139072832220/11621486809, -529457353983198716737955152173651410168139811897740929085970/1252831142470627)
(20111189058767673493696094112230871635120/11020170529, -374588826087761624891903798792444281552666677594101222296630/1156864441622833)
(668723586210295069272435882381918200/380689, -60504629242600300525729501622692413388321949687474930/234885113)
(118696986067578944643893158595302220/58081, 10670843495112820204460659359993339198505120443280490/13997521)

Commentary

Fiber t = 44953/15632 of Elkies' rank-17 elliptic K3 fibration (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b and an iterated 'bootstrap' 2-covering point search (fake 2-descent over random cosets of the growing known lattice); 17 points are specializations of the generic sections, 11 were found on the coverings. Search run by Claude (Anthropic) with Seth Lupo, 2026-09-02.

last edited by Mundoamundo at 2026-09-02 20:40:40 UTC · history