curve #618

y² + xy = x³ − 6970137122541813902149900448031453492885543816565287857980x + 223354420323646757129681518060226271626563374061988532595018210711139585894230496167952

a-invariants: [1, 0, 0, -6970137122541813902149900448031453492885543816565287857980, 223354420323646757129681518060226271626563374061988532595018210711139585894230496167952]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 56028086190794798833781577518120957209584777040976670351674315922352499192998038665361259294688724967685308936002906025771359335350230
naive height: 411.1806
Faltings height: 32.0775
discriminant (Δ): 120977774424433983462971056824550202872761785070229185596995550243815010071392850241810257134585388769459866768131603105216084679013869650019019416785449115209999187148800000
primes of bad reduction: 2, 3, 5, 7, 13, 31, 37, 41, 43, 47, 59, 71, 223, 569, 1621, 250621917391250283990175327445045688280396652657390602160713828693719061401002824811135041300877175441058588951
regulator: 2102485225710153985861312449786776.532123788966123819948767359974282353355738230881982570511165
submitted by: Mundoamundo
submitted at: 2026-09-06 22:22:58 UTC
last updated: 2026-09-06 22:22:58 UTC

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

(70563345632463359038531122944, -9103089050593186055947857149527405040007172)
(13631225660021850140506109944, -11440093121621456140849371996525829212379772)
(3336719583851106144392185443496/9, -5952226751083578413502302402473403012908362644/27)
(-41582192243328345582975197116, 21006882651151349535912777254821969344541748)
(50669616988495156962744142568, 519545633280057490507150032434861462996996)
(2206061514321536904874072949464/81, -5343257397029103330756170476579839875454379388/729)
(51123365361416037770258139314, 795929576217388026449940809166916113971198)
(23815951246905642669826518299768/529, -10914936518374245012893167759100594806409278820/12167)
(23084902693853351797020899768, 8645907798396278596789825846610651569766276)
(51482003291290553619408183364, 982612722309965318962723476263396341134388)
(71165479169025842201614805144, -9366997960039983038383972832956075755826972)
(413538198681903615615184903096/9, -8382990088672450065817606606109987153639444/27)
(4195746916385919968330609494136/49, -5470237536756761433367314708126088371708974116/343)
(57614049961426664758157809784, -3608246834868033837130263220615938120244732)
(72154069210741524384444903224, -9802034444927039342950338042974127311727292)
(50563827344602931988030420644, -440658380570210824178380881907103133684172)
(240278931361615421389686669944, -111448789585223379533037219127622330273970172)
(237089293789737029337711178543448/5329, 445988445531745862126714171872189436521930408676/389017)
(164599242115756130663309245724, 59460471477753587615594588042043726977140208)
(473012934558215006778515421044648/9409, 50356987502725914755166854830650063984938639060/912673)
(45750408157335753321766893944, -477466594712753584319187970732820110866172)
(50574538150162558064872053704, -449103995256219967940794892011584182353612)
(-796999100692252113224801470064/9, 326398755584177018830611854610733919109362276/27)
(-93451777240749354235079521768, 7654398808256498744943135625218495861006980)
(268208648208371509812066602426503736/7745089, 104511486247782949214754503204423881209184366647852796/21554582687)
(19439042455032789771176458544, 9757416856465203009787031946630151024318028)
(169050407572792855456655389281024632/2686321, -25649768579818112138571125301221385634628022909099940/4402880119)
(3174220058652917834506978757624, -5653361998430021080906570439736973186226880892)

Commentary

Fiber t = 17308/3257 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 22:22:58 UTC · history