curve #579

y² = x³ + x² − 219738272640944325683337764571119491754140699793033018904940x + 32559411551889641574478224228029372349256698844813794220096983309463248510389444089600400

a-invariants

[0, 1, 0, -219738272640944325683337764571119491754140699793033018904940, 32559411551889641574478224228029372349256698844813794220096983309463248510389444089600400]

rank (lower bound)

≥ 29

torsion subgroup

trivial

conductor (N)

2134493204241013843464057484549643880507283880975565758791441628605126595168796467420348337493730544916532571757798781831632744990694528986633693560

naive height

421.5330

Faltings height

33.1308

discriminant (Δ)

221072911119964769465452171614686123031021954267318901636001618436080142762181570882207586051638153243689850928015916640589000241693839296967056618088437749695799879912326100000000

primes of bad reduction

2, 3, 5, 7, 13, 17, 19, 23, 71, 139, 19403, 413711, 1008442274129, 329345425514116035278489823097313159176453856247834130597191055151101727667566724597818136056396414775659294913799

regulator

2262245440824371339373802554427439368.21923951429628186218174702400746322737076840203045520188107

submitted by

Mundoamundo

submitted at

2026-09-04 18:47:58 UTC

later contributions

primes of bad reduction recorded · TristanPhillips · 2026-09-05 11:24:00 UTC

Witness: 29 independent points log in to add more points →

(55028932304450889534514044320, -143645698581054502294228976595038666220234300)
(704334989786168357709844422620, 476657148767983829926449074837268097519804500)
(132062143014908229982690968995, -76442934491699906716140580887530584136812500)
(-444644737681865557632277281305, 205802422637264703878733952919534406164124450)
(165630930151265690297084183234, 26604709768422900330137226652051990834493490)
(454499210257836288730007689760, -163016040276106435480908005298008497895590140)
(-425145506643136247527260826630, 221665631961402167230464463083923253285266250)
(28801651041190775468706009801635, 154549941082822294293199852277101204501526419860)
(91624295828709612852282855335, -114870510051958128699115226862600296877219090)
(2709113277829138516994223004169/16, -922272916399987204924473717283708519552800045/64)
(5783403022667240377742851501625/16, -1213018426237610517118836426599813887609980405/64)
(599629740160660360564442519270, -341171766583679635813680099383158853841410850)
(7752675626374039232085788012570, -21547520872566693500515286291166146737595731050)
(169948540678325469537298776995, 11124415156366848655267964613336133537915500)
(85389382588584902617276453370, -120077890177320493009318009138123463233106250)
(115760392043963555162917916768, -93132520826784208207473062830197411471444444)
(12863990754876088597049027393524070/101761, 2677183559649201883672777134020817824651802581129250/32461759)
(70042366145682273147239271275570, 586180689647311742916768568247909855797326121950)
(20947848282693056381087135550005/4, 95501882609961729997626205952363919049036426425/8)
(1073873867251527579036706140890, 1017342137577896503313822420101597183839045930)
(720955946957046735041750395223630/1849, 6240556921837819230977863395404737432565336261250/79507)
(-56128503151253639296298677165, 211461968557716973329162585633186022359567840)
(671339940927170294641839672098, 433141365223732895759003565444112734112593774)
(1077052352017498550923531888220, 1022406721369273675837814407165217046257745900)
(-398612331461484009949220685655045/2809, 36734896196425805453219794485301932933243093681000/148877)
(-1553787447641171017144440773023630/3481, -41944279337802678856226255100813101106221308684650/205379)
(-3487087104163453630099869288320/9, -6587835573117649277906812510957585695892939300/27)
(6589588993528936136239787043365/4, -16280469235956240603609465606176915443251297855/8)
(-113729837695068007534036490080, 236810417381960974508082554111264203874405700)

Commentary

Fiber t = 6150/32059 of a rank-17 elliptic fibration of Elkies' K3 surface (arXiv:2608.25406): the 2-neighbour fibration that also carries NDElkies' #404 and wgxli's #393. Found by a Mestre-Nagao sieve over t = a/b, re-scored at 2^20, followed by an iterated 'bootstrap' 2-covering point search (fake 2-descent over cosets of the growing known lattice, minimal reduced quartic models, ratpoints). Basis LLL-reduced with respect to the canonical height. The 29 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 2026-09-04 18:47:58 UTC · history