curve #6128

y² = x³ − x² − 193326062935853369206366692096471105x + 277226613981010057892227345254004285684975924574892897

a-invariants: [0, -1, 0, -193326062935853369206366692096471105, 277226613981010057892227345254004285684975924574892897]
c-invariants: [9279651020920961721905601220630613056, -239523794479592689963206520173933932500385591508923784704]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 1267548066931185496991495042988782951726511449140210157873613120 ☆ record for ℤ/2ℤ torsion, rank ≥ 18
discriminant (Δ): -32738749726314483519098577760507579917483485715113725033745485280593228446093741552814609460728332605546496000
Faltings height: 19.6941
naive height: 259.6365
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: 159550392406753899890880658.83534071534382214987383002562103386508713457210
submitted by: The Wren
submitted at: 2026-10-29 12:22:58 UTC
last updated: 2026-10-29 12:22:58 UTC

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

(-15956347108569918990531/21316, 155300493675335707661084677425465/3112136)
(-7718381982982044169133/84681, 13363474597714012121944525833363764/24642171)
(38013338251026425778875976/101465329, -518520500821815754687204876617326124915/1022060259017)
(92786865409177196512, -893768314581545962284366821439)
(1771717478958853527, -2344375862950167904396832496)
(30530554490753756558503/124609, -21753226148020296067945152033456888/43986977)
(67681908070903796931903/34969, 17489832070880244516339642826167792/6539203)
(9680200480396703703205128/3806401, -29921061661045812538474190151964027995/7426288351)
(621661089169200856447189685451/125185054225, 488783329489795035835577190274639397528073876/44292349960618375)
(143972858326477674115272537998499/340316122580281, -3269138923661414735668528070650580911555155645420/6278035781563224012179)
(17796569965681257848765912372815321/1318486687584784, -2373003723798585576027608255940853383979961701772715/47875533195263839450048)
(226468233355497003049054975797/17037036676, 107720628913208710651367265834139993071999099/2223776249171576)
(2616885680714259103382743551/10781707225, 553683091700074974499793458428901470925776/1119518569707875)
(61783109755595402484564291602919/41545256604721, 484869921937955772333810180742404621870031499760/267782485706382093481)
(127399426537511011042108286810584/50956968712225, -1428460245807701711882516419029051171645197703027/363751989809877623375)
(929896114993377339067507483/1473254689, -36060677782269862351953098598814638759588/56547934727887)
(-17461778865254827573227814673997/62251042744489, 273192789977352797270638200521177936994470420652/491156556434154329237)
(328492226040371378934855536026479/10057403738281, 5953193745079677840675253914259858829494387187040/31895456828763804821)

Commentary

Rational 2-isogenous partner of curve#6087. 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 . Parameter T=30/769 in 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 independent source points were transported through the rational degree-2 isogeny and the exact minimal-model change. Their independence on this endpoint is certified by an exact rank-18 F_5 reduction matrix at eighteen good primes.

last edited by The Wren at 2026-10-29 12:22:58 UTC · history