curve #6087

y² = x³ − x² − 479337962456030241119664973800901105x + 127023999849553558599285116689332611801173040541538897

a-invariants: [0, -1, 0, -479337962456030241119664973800901105, 127023999849553558599285116689332611801173040541538897]
c-invariants: [23008222197889451573743918742443253056, -109748735870014274491733007632246667153749994573230088704]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 1267548066931185496991495042988782951726511449140210157873613120 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
discriminant (Δ): 78280247831823416883598446566167686668640237781810624016601302065529866200279962668787687229670123776000000 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
Faltings height: 19.3475 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
naive height: 258.0867 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
primes of bad reduction: 2, 3, 5, 7, 17, 23, 29, 31, 43, 53, 61, 113, 127, 149, 271, 373, 433, 769, 827, 863, 1033, 1453, 1657, 2053, 6991
regulator: 608636445643439864696.04743513237272393730983686000833748258157763341740456
submitted by: The Wren
submitted at: 2026-10-29 01:30:53 UTC
last updated: 2026-10-29 01:30:53 UTC

Witness: 18 independent points log in to add more points →

(2258983854455671645861/5329, 775515975451240293876447955680/389017)
(-16259614282986063977381/84681, 11345646119852791598566786157298500/24642171)
(6649760696320392856/49, -87092757280339230083992365885/343)
(374496806071164549, -5986540705489264141466000)
(1811540234946930799, -2281134755165392558287534000)
(62204067856959549, -312166490468264236365316000)
(11864257234598248588981/34969, 384043582713796052296516051848000/6539203)
(1328604131142040377566149/3806401, -351359047261492192486914840700770000/7426288351)
(15059549584431863701459/3025, 1831069915435756430061215947024548/166375)
(19530944830979009070502501/123054649, -319965281520484122258514606858649228400/1365045221357)
(400241233501201929364149/1079521, -18448992523719271069218438054502720/1121622319)
(12962625638690360186893/34969, 108477005066850132231214113111072/6539203)
(548292344516115306799/9025, 268574954335610483796405808415568/857375)
(154389926241967830667111/100489, 55332345305337478226120762555746800/31855013)
(243003771145938471080109/697225, -27904664342130142271990831374940048/582182875)
(774434750215117197, -469334244269555516353513424)
(706822451155086558406477/1560001, 104127674475601395333181424201626704/1948441249)
(90333173155162165286269231/2765569, 858369167222904725305249916411286832800/4599141247)

Commentary

Original specialization/search by kaankolcu (2026), public Yukon elliptic-rank challenge submission ddb44e4a-f316-457b-a3f2-7316d2cc04d4. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/236fa93f21f753f95316d5a2449371577693fbb6/submission/board/2-18-own.ec . Specialization T=30/769 of the rank-nine X808 fibration posted by wgxli with curve#4078: y^2+84(35397T^2-221T+20)xy=x(x+F)(x+G), F=(1139T+115)(1269T-59)(3776T+34)(539T-5), G=8(638T-23)(250T-1)(7811T+499). The eighteen supplied rational points were checked exactly on the minimal model and certified independent modulo torsion by an exact rank-18 F_5 reduction matrix at eighteen good primes.

last edited by The Wren at 2026-10-29 01:30:53 UTC · history