curve #813

y² = x³ + x² − 181407490007347578063914880459645863566134327545x + 29688398854301875822513612006047654861357374193053111118868029793923768

a-invariants: [0, 1, 0, -181407490007347578063914880459645863566134327545, 29688398854301875822513612006047654861357374193053111118868029793923768]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 8091264367067128246845265990756801896257312709055258682850239059670459161970309617873949080448232068224411710427911077820
discriminant (Δ): 1307113157435490651616797359949414936100640212932186928082253090665649592901372329607820694015233631167928728919662384919685650640518775250000
Faltings height: 25.9655
naive height: 338.0648
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 41, 487, 112859, 141199, 976453, 171590737400253477740022610062342199466245899522264992745535123858798525188424659271771284721
regulator: 869636371320496661864192910218827560.729095436221089250342711568358565078480151349190572365
submitted by: hdeping
submitted at: 2026-09-21 03:34:55 UTC
last updated: 2026-09-21 03:34:55 UTC

Witness: 26 independent points log in to add more points →

(267478426753353499933231, 17391807624518293936540802744885625)
(1198760292814853003119790449/4624, -2861058197173540993733611255003812137295/314432)
(278656285557244705236601, -27848413438290784085099067622528335)
(444957547187811957315361, 192524477172773939575597444275581535)
(31642337349110974293948781/5476, 68577925036258733026255597664710261571475/405224)
(695788806053902939190431, 490218124832687193625273920070328775)
(909636308409440333989591, 785710761897186159010778130901042905)
(-2742129596304657838790591/9, 6427821375775309131029380318311578795/27)
(81410933848148686739023, 124335900691118281698477214915836423)
(237520934093390478590131, 591189130239737468259435610966425)
(-109490045518988516746019, 219631825688557772750140881411035625)
(271044601551727996378381, -20765774048442770763804058879699575)
(27040610891896372604436136, 140595426651833660560123590109591842540)
(68793991602271373144530609/64, -531263547573300240195434871897631661175/512)
(40571685807425147302898349/4, 258200769965301794022106773640143887325/8)
(22979818326907960827031549/4, -109864464041784477376132917093530656875/8)
(44024419986976401640060231, 292092371252849187396457341762632105625)
(192878469707183092024767259095395836192238349496341605980879/2039521603572947785863914863210128609, 336895101279418350259767526988119821030214825214219246720168927242805106535594663127738825/2912677928783145889089278025427786579676890202854823727)
(-2302563117286779781248790321789747539339033827016760323541409/5616522697927904475962779090704788401, 2495771196669640268847848595684804065736912052160241286694078229789900462402373216456048545/13310711713658771202496149943684196777533102325703388649)
(379114883847559970712882709743669537039722693083967830352893/6076162534911831164085922054023596356, 2043372414684928804576351489498040346607332650680260727145124873858899297373990519883381173/14977663660930090139759376904390601140150480596056354504)
(-2711745346801238642005110474568984462113606885216412385121891/5832733628685552264917795405926454689, 1638925531024068264774356585465525081679445860786711042748024040197663542781252172089715175/14086665922584124203409954917107712341252853218698127887)
(401006956043219344079767931243889031554389754140939013509070229/8039701508211930020335393944686323684, -8029935868683435363105160746517832058625937904530833130806990419956403218446840106966531772975/22796065037385252967427763208017471690822446105552646248)
(348818341972311301593831991713723287228328379484983753182069271/3480452003769950304118482874505372930041, 22969529090447080930602716443884682485214528238719594944142852363490115167805465703961666845275/205330504242376286631700799465237482285228674547812607495411)
(58752872196192081578764884167030393131479093378182765760400939/84733968672773307488182591072735634169, -379929551133505525602473035972929323147985514946376069890350442294399488270625423427141540475/779985127540722226129849329001153075612366260996675634803)
(-183406649938624078125229727632034552211281727611366064461222876/1152927679452674340763271331237068181649, 9140804530896005179762784865136788992132344408483576996977203118739383018245907003496094598270/39147415806558643490620939692747153673475865283533511281607)
(-77575566064215305495607607293078811591637855723447240655348449/1157046826410043506050527498934654147281, 8022510768066110973322335010963089792418892580932219647061626955224820878129728678822167245905/39357400231620271092955272362830045098310548045393635147929)

Commentary

Specialization at T = -511/4961 of the genus-one fibration with 17 independent sections behind the rank-30/31 records #273/#302 (the 'wgxli' family y^2-(Lx+B)y = x(x+D)(x+E)), with 26 independent rational points.

last edited by hdeping at 2026-09-27 08:23:06 UTC · history