curve #534

y² + xy = x³ − 449002837340028581084963490553612776357829051073257486319285x + 115734052939476590132424578134748477912680735992109057724697743331116312612652932287608225

a-invariants

[1, 0, 0, -449002837340028581084963490553612776357829051073257486319285, 115734052939476590132424578134748477912680735992109057724697743331116312612652932287608225]

rank (lower bound)

≥ 28

torsion subgroup

trivial

conductor (N)

110965370156935641841165491853870644635668940806652922944505209680823632292021535619244391555545045101884792106951752370391869388854751151799352825370

naive height

423.6767

Faltings height

33.0615

discriminant (Δ)

6947886494192190796439270488064223472975765288881477760106172381166513407895720627681112520685792655765585287305870098574336869539490502779069506357357362592104548552859648000000

primes of bad reduction

2, 3, 5, 7, 13, 31, 193, 34039, 224784925918951, 6195562674700620799, 143311514199741708869365285504280830844923716905413759918639517739619223658828328784601167078556859655513

regulator

568509901552819130144647074202448789.1028141325479177700380094541641741671292010270913635558014

submitted by

Mundoamundo

submitted at

2026-09-02 20:38:37 UTC

later contributions

primes of bad reduction recorded · David Renshaw · 2026-09-03 03:25:50 UTC

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

(-750244334455698535452968187230, -174093190780985299664664045316898748335936585)
(-329612927892421751667661950830, -477410202002532165242781036032586522179180585)
(10455844713366335568772840817362, 33741710066041590912556547559273053825473150231)
(-569373639567399657783552960218, -432205131264555847414624604800637439780523889)
(2911065427140792077447241597730/9, 1779371906197864148779917842303049421431756205/27)
(40756746726817247287040551570, 312252877578271941352031892663086975544531415)
(502892474890357365398680995070, -130827500246109880923040294536804604230721085)
(-767353790081360392484306930390, 91846276111727217025138874206786878014670415)
(192153721868405006557658862415570, -2663609555056572949037535178504010221595628586985)
(2795349122347806262332905264170, 4550099704428116844858190221504312987356334415)
(2454984856555249224476057305570, -3716115102314249604772092746386882196746178585)
(1163473028864999885999780723470, 1080874323008370963241485830130716030239963615)
(10862267373553101264784898190970, -35733247936296745695465092541774457524104116385)
(378549836327928947577955604170, 3207147075644211183971861499100363516674415)
(-421062116650473404720997558830, 479729760684804759578518243399196940817275415)
(357609233757347907701968601170, -29985359219494402289262652835890250168564585)
(394589480048661223948645499920, 421095893757633460458614033574807042821665)
(1122199742535976634183212898985370/8281, -180505144705133136090295342947643350602119875450035/753571)
(354867125736680542254720264082, 32960035933502151623917980685718331897980311)
(34543896629031202297472278251250/81, 31036447574679047231766014576939275928943902335/729)
(3129262543658147741468663787298178417705170/2173386526081, -5019350924345533373290190070235371091630003861756470217621958185/3204095525596179521)
(-171948688793917010782105336969535/256, 1384751289460949802368889297118970289641696165815/4096)
(4167342223812769452997269239150319538417270/10992533619001, 1592591050928331605248933222882730806829437748410138637543185/36445734221264196499)
(-771995834426464190600513097198, -47646330572135584814261359724682092675714409)
(570993944050575914442514121170, 213353320940169694679831708756980161734523415)
(4717781798909303217218270575510, 10149054162231979774668499078620240073879393915)
(1824199560261580922069985557344114/5329, 18020827768131648369338140472015978726495977680319/389017)
(5775278657215261159699217274570845890/76729, 13878491510989706463009151241275141421799013378732093195/21253933)

Commentary

Fiber t = -41395/19353 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 20:38:37 UTC · history