curve #537

y² + xy + y = x³ − x² − 191914065213833161454919231365113968096632867810795672x + 25277597117582435443530894665224009261825515258125542567000517543470385786814619

a-invariants

[1, -1, 1, -191914065213833161454919231365113968096632867810795672, 25277597117582435443530894665224009261825515258125542567000517543470385786814619]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

22883080138526996222853278176402454461389320384421571738020060344839345277376684497478977149200990523395796406130486199688704413673890

naive height

379.6803

Faltings height

29.6531

discriminant (Δ)

176347479671630824953269265517657145803426562107881664722784499730209567228150394803364507109527791690571706531764007346125995957226029971767514975393507328000000

primes of bad reduction

2, 3, 5, 13, 17, 23, 41, 47, 79, 109, 241, 115057, 71733731582368692402489696151363, 1299892134541896708944316221148973741, 1165885042422770955835658260747122788076534901

regulator

735923745881476085208359244966247.00382232280545584872836064233873991269348823061946616099431

submitted by

Mundoamundo

submitted at

2026-09-02 21:24:05 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-11 19:56:07 UTC

Witness: 27 independent points log in to add more points →

(347365269116276583583668517, 726170238298363845011151411474917040761)
(-485719894450099505988209723, 1975141938370519031963524316087359416761)
(8301324117481627564306691327, 755309761460929500130130042559583004312711)
(-269962563662925726264095423, -7577096829093122658803121164401789529539)
(802316973161499678749849879, 19691703080019162601944915572910834519487)
(-182239119559614701839661081, -7362030044181336341045617138496854965553)
(148828300599048685958799157, -108992698632327131352655624231847187919)
(349114258737343041387875677, -909930576871184174578039375770580855639)
(134457838587513910881819680653, -49303401826151947965309808474911659222496247)
(382185928624017896040513877, -2784805892057605730789842062744218924839)
(66933070952737097426630989, -3568201499410681786639646870524018116823)
(4106989128024727001453745037, 261746349572939155301421194749274848228361)
(23371101516883026175345844677, -3572258623854595649449781660352721287469639)
(16129594009909247643523592923/4, 2036768604469993130092004709418317510616633/8)
(344206271464053928286382877, 20693645331928124713372156358409861161)
(144408213542851592305082527, -758340299699559518092097489203916230489)
(1168027058920508314393024237, -37344894615092236966866525509911882892599)
(3843356869378074805729686486791473/9193024, 118638168834291745505729918051033284289821678829923/27873248768)
(-151538152951595495996634818342027/368449, -1316250925130889704050774065813541333062454573177/223648543)
(53738544109404146514505737739942753/384042409, 8116746131987927577158695767911930286769156278616133/7526079089173)
(-2826572560301978590611094188967/12769, 10885325644769703899166804767020324308437496257/1442897)
(520810250373731832055483689483133853/5147919001, 970035132588677471943763798060482901950506424990197917/369358040402749)
(115211664393911120815263888665609693/43681, 39106114723300370888928483078360192422035465104324697/9129329)
(-91700310951457498828994749022483/261121, -937492416344771444602004923128643425784651662409/133432831)
(2327046515470116087080119141/25, -358389665644486537919911562326180211503451/125)
(32249578843214314488543264764227/78961, 86004366641305361100402311869705677148058177971/22188041)
(-353409547188211963705771362707/1089, 262562697629862065466965719361762076788609257/35937)

Commentary

Fiber t = -2602/3983 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, 10 were found on the coverings. Search run by Claude (Anthropic) with Seth Lupo, 2026-09-02.

last edited by Mundoamundo at 2026-09-02 21:24:05 UTC · history