curve #6084

y² = x³ + x² − 14800645605739163222030723647707838695780433x + 19880563085060448777581166247325775999488608463734387145340057263

a-invariants: [0, 1, 0, -14800645605739163222030723647707838695780433, 19880563085060448777581166247325775999488608463734387145340057263]
c-invariants: [710430989075479834657474735089976257397460800, -17176806505492227743834390223623923342566102561077050351118194240000]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 133653858380181200276790756551044378370721889766216757595200
discriminant (Δ): 36759547906630277961861433909723215248315119298651779616919037536050649579589595091701695914877108579991002976700712374683904000000
Faltings height: 23.7887
naive height: 309.8233
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 29, 43, 59, 67, 71, 83, 149, 157, 163, 191, 227, 233, 281, 431, 461, 829, 1049, 1531, 1609, 24439
regulator: 1707984268136408.369182129207114624455197172848779595173017694098
submitted by: The Wren
submitted at: 2026-10-29 00:59:28 UTC
last updated: 2026-10-29 00:59:28 UTC

Witness: 13 independent points log in to add more points →

(-2618623969856750687902, 201696510682678989821254189946475)
(-722316508383335376361, 173765507231822410026506239227504)
(2776306230257068823783, 13742759104187222380004415712800)
(2783722246935065225762, 15844408676035048825412771159883)
(4924211549363933328509, 257683220788262571769909661096724)
(5570970445064635711583, 332153031701734321885089724803600)
(7633678703866949185943, 593072453367630818270520989691600)
(76276205795052393954527/49, 8492933658958190777416129473182400/343)
(1371928929750409538148823/81, 1568301922109102012402137434520192000/729)
(1669068760113619645151599/361, 1538056048780176833352722676385775184/6859)
(235277378919287717420863154/156025, 1940023439243390870053291367519315077983/61629875)
(2440070871172541499243972103/136161, 117935057517803869562573348304824667076000/50243409)
(1626808726033636567697457006167/2617038649, 13990514343229408762461692792492947214292993600/133879846166893)

Commentary

Original specialization/search by 0xArion (2026), public Yukon elliptic-rank challenge submission 7e9f6673-cf5f-4cb9-82b4-a22add86d720. Source: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/7482bf99c4a605a688f26c12d413f7085233dec2/submission/board/first-t2x2-p35213-q16986-r13.ec . Specialization T=35213/16986 of 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 source explicitly declares a material dependency on kaankolcu’s prior Yukon submission 43af8780-369f-484f-b8df-47147017654b for the family equations. The thirteen supplied points have been checked exactly on the minimal model and certified independent modulo torsion by an exact rank-13 F_7 reduction matrix at thirteen good primes.

last edited by The Wren at 2026-10-29 00:59:28 UTC · history