curve #619

y² + xy = x³ − 138217506563605889872043263854781697377016668744941773513580x + 20242130248207841544219808693189001887421294913221456339098328387159912218373571988160400

a-invariants

[1, 0, 0, -138217506563605889872043263854781697377016668744941773513580, 20242130248207841544219808693189001887421294913221456339098328387159912218373571988160400]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

3389128510548161319756016765334023590717644951355796487807190277570543774230467803189406795413101608614781556190293416559648058753190270430

naive height

420.1885

Faltings height

32.9177

discriminant (Δ)

-8016173977599554153084600438157612362508330895282880069000108443445706401713332279338193539308317679859440802108986091182605797989242654639201392845727839789311853979504640000000

primes of bad reduction

2, 3, 5, 7, 11, 13, 17, 19, 37, 41, 107, 139, 241, 2309, 146750759902520012671, 174361184822160214729, 1087614221970164994197309267684770841847552381021820065017808837975906824362877

regulator

7193958602848888263120749698399987.587458367269755059106587452516299766720719481925179014864778

submitted by

Mundoamundo

submitted at

2026-09-06 22:22:59 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 20:14:42 UTC

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

(-285556687979304397662486502440, -190856025740885428017626054283636023707092780)
(212608797143323944978730729560, -21594483527959644179995745821414611075492780)
(162621302610822882023270839110, 45449439253228940905138411472973023458848720)
(112812713504523566044822124760, -78007517454595962910958646652315224156967980)
(473206610682428914467707394810, -246575027968020494416211842514958749868173280)
(205709387537445953009642216460, -22679561364624694871980113709186383247705680)
(2319473965629430687212421739490, 3489745859767199551318356787725955152982350270)
(512248397649888795436013086560, 289574868787158822975795028606585724833622220)
(7244663259205224489434963784164760/3481, 607418670414168533998243774450500991894017634864380/205379)
(400638962258680685051619721560, -170803992707845388992087678795251996894852780)
(4713476717579976401069704492560, -10202324368584688136994899409192226986850692780)
(88405210028393264310358999800, -93348335841024634500707638068961251067867980)
(-13756267996900595825298707940, 148798132193367312751804080852037793464194720)
(-260657820576430130997987559770, -196366648006370902912640117962210589337553200)
(2094305227967751936719195036460, 2986073471414322224487664922815428936591089220)
(311893958097606216910003498200, 86448226950487746166391329331340163227736020)
(-404527437742129256681980658040, -99784988250888031933914848582821435244121180)
(35194723605693433106982361560, -124182115883015161827766226736200394825108780)
(1282717526106673310133114871182, -1397671518903307108749869981295959980299634224)
(8881763898544446339459764990040/49, 11602007189397380305574026338646026568670312460/343)
(644599585747118925243325146360, 446076082000637451826051817740708973855562420)
(657709991643297813546665701560, -462437978139149621974285629490754724023652780)
(190906527013919742028636890840, -28515822611473367570735023984690579252711980)
(3498132828906465552627461585085/16, 1393328771514670586697526547749314816615465705/64)
(40060815475411013615924206977946560/22801, -7851983511048995078590689460549005728845612201353780/3442951)
(867108604640879705984909316629400/5041, 14120991689567452211853706379792312654760300106380/357911)
(138270086583894370878645248870375640/508369, -19075489909815272193555641849848801197627267693651660/362467097)
(212580295368926209015741878422739798552/1673546281, 4710912104989963491693186180069183451410902833378019389668/68463104809429)

Commentary

Fiber t = 19348/5487 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406). Found by a Mestre-Nagao sieve over t = a/b re-scored at 2^20, then an iterated 2-covering point search (fake 2-descent over 3000 cosets of the growing known lattice at level 0, minimal reduced quartic models, ratpoints). Basis LLL-reduced with respect to the canonical height. The 28 points are certified independent by the F2-rank of their images under the 2-descent map at the good primes.

last edited by Mundoamundo at 2026-09-06 22:22:59 UTC · history