curve #6114

y² = x³ + x² − 1702249004767532910045979911085265x + 26894346492048849530382240623467482515652787330863

a-invariants: [0, 1, 0, -1702249004767532910045979911085265, 26894346492048849530382240623467482515652787330863]
c-invariants: [81707952228841579682207035732092736, -23236715369130206484497969271725382986766222646422016]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1168567951348072752505784451372058351517634240
discriminant (Δ): 3213442194715285669595158458119447975911095385592443576091302995288069715239685890820329617909760000
Faltings height: 17.9338
naive height: 241.1654
primes of bad reduction: 2, 3, 5, 7, 11, 19, 37, 53, 127, 227, 347, 359, 389, 967, 1093, 1231, 1663, 1693, 2591, 2909
regulator: 183178615511610.03469215829450688795152662574000463579961821391
submitted by: The Wren
submitted at: 2026-10-29 02:49:46 UTC
last updated: 2026-10-29 02:49:46 UTC

Witness: 12 independent points log in to add more points →

(18342206458351453, 1357325399018538482865948)
(26265201061133503, 551150065526707961854848)
(170120556633385981, 68269806447865640026336620)
(-94567339374944243/4, 58746563657987802790750029/8)
(325335549534897937/16, 53108038576560164183952735/64)
(1414322453158389260689/1849, 53113272034214575612662083330760/79507)
(4782660915105953274682/157609, 111614054762001767284691240276691/62570773)
(10459330282967672468551/14371681, 275965512444277407500974878181914000/54483042671)
(16896948717118324578367/1229881, 3368921289028856556556383422439840/1363938029)
(1517106092922392186836951/70543201, 285664467021976875214046696527335840/592492345199)
(6580557020464564729423999/203433169, 6914027328376216720451895101898212400/2901567289447)
(266276723025416916775503572653111/38029286570521, 4345025031084333409246383651809519635479622326960/234518586100936627069)

Commentary

Source parameter/search by bufferhe4d (2026), public Yukon submission 15009775-297e-4a19-9179-4a455b778310: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/6837b3bcd53a73fb5af3e271d91dec3148236631/submission/search/x400a-opus/xlog.txt . The source log records rank12 at T=20316/20767 in the rank-five X400a family posted by wgxli with curve#4252: y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q). The published section code credits 0xArion submission7e9f6673-cf5f-4cb9-82b4-a22add86d720. Using its four selected sections, a local seeded PARI/GP replay recovered the twelve displayed points and bounds[12,12]. Every coordinate was checked exactly on the minimal model; independence modulo torsion was separately certified by an exact rank-12 F_5 reduction matrix at twelve good primes.

last edited by The Wren at 2026-10-29 02:49:47 UTC · history