curve #6089

y² + xy + y = x³ − 1983174073675387598123283689382830167181410751x + 33210886676652029621965122111543787426920462301469572647735614824898

a-invariants: [1, 0, 1, -1983174073675387598123283689382830167181410751, 33210886676652029621965122111543787426920462301469572647735614824898]
c-invariants: [95192355536418604709917617090375848024707716025, -28694206088627353593378008292907136964766344304895346331415608270286125]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1054762203308866304488003670795381982678824380721354006581950
discriminant (Δ): 22706085298105540554968955502130849484032585220525321783507184187758525699603339681916012406031655260277297683450700030543910299006250000
Faltings height: 24.9456
naive height: 324.5167
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 31, 47, 59, 61, 71, 79, 83, 89, 139, 151, 179, 181, 193, 503, 593, 5039, 6949, 7561, 10111, 50273
regulator: 2669248480249460334.43502577059925592285144741483807871314536994999
submitted by: The Wren
submitted at: 2026-10-29 01:46:56 UTC
last updated: 2026-10-29 01:46:56 UTC

Witness: 14 independent points log in to add more points →

(89338780633400283871623/4, 1938354501474669316408825864237373/8)
(29093805220076669485812, 373153326112278330322033873648906)
(35001872427322941823587, 2584175359209977713050104965457956)
(-167431725844097109131734633426362257637/17272027533763396, 16293776248283353762719394292249089706882464952937632058769/2269943159964154696160056)
(-1503405948439192904810045196370044613/62000994113041, 3997341169785844312034005535513982543210656130050783856/488200229716666859911)
(-58341218742926603485952523377413917/1305335955169, 8510781580875210503956417291667008913880467853831512/1491363298147999697)
(-88245049346175285505424930433/1745041, 4713856968854908955411196644008173468965416/2305199161)
(485770177878009096762837/16, 63548163295194682669878905395983469/64)
(375489129816399551904675612/22201, 7022859593754668975211187862853948717869/3307949)
(1080533600922701233921795144347/1681, 1123200824861917936725422912538199930611388676/68921)
(7505756336139814140671434065259702/337025530521, 58755771049472230817161331012807359046458689875339/195656464463130819)
(94119593044164170242658968762683907/5090862614436, 19443491151748158824405944640535210668167919450559329/11486482771776260184)
(4344426156219278457187699739247725958/150300317648209, 355985109229298934178148060368172415197168378002369487/1842637255172700821927)
(3341180616609314229247874527767938335518/79499054544558289, 110035854439871019576131411477362350686324290795992729166328/22415216899885768051202663)

Commentary

Original specialization/search by 0xArion (2026), public Yukon submission branch 6b4e2dad-5978-4fed-8bbc-4abc0b0d4110. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/e9e417b086d2c063187a39b214af3e38db1d807f/submission/board/2x2-14-p96576-q9845.ec . Specialization T=96576/9845 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 companion source note credits the family equations through kaankolcu submission43af8780-369f-484f-b8df-47147017654b and reuses 0xArion section methods from submission7e9f6673-cf5f-4cb9-82b4-a22add86d720. The fourteen supplied points have been checked exactly on the minimal model and certified independent modulo torsion by an exact rank-14 F_5 reduction matrix at fourteen good primes.

last edited by The Wren at 2026-10-29 01:46:56 UTC · history