curve #5866

y² + xy + y = x³ − 5637986040127919329120761481632247473x + 5152409384201596055321791465774648975603994977863872756

a-invariants: [1, 0, 1, -5637986040127919329120761481632247473, 5152409384201596055321791465774648975603994977863872756]
c-invariants: [270623329926140127797796551118347878681, -4451681707950178992203962821318506906618546487551907879421]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 581459793254966336566732749789261962353630126730
discriminant (Δ): 1254158389340923198127704514778979327874655786743780209366406136942795241891565868290465098674580566406250000
Faltings height: 19.8005
naive height: 265.4814
primes of bad reduction: 2, 5, 7, 23, 29, 31, 37, 41, 43, 61, 101, 181, 1249, 1373, 3727, 85247677, 166167293059
regulator: 5148217095823413.0356775419879002538795452750984118211688952248
submitted by: Maksym Voznyy
submitted at: 2026-10-25 16:41:21 UTC
last updated: 2026-10-25 16:41:21 UTC

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

(1711609172912887431468355495/294774561, -65633842731339752707412393028704484684482/5060984437809)
(2316521638686697795, 2126729670137785265736651102)
(2880683613961887161907708295/2054446276, -5720599214592964269183906628307605980073/93119831905976)
(1852142070274200397984165195/295461721, 74665576663727131559631987195543145790438/5078691522269)
(1549960160952537625043455/94249, 1910569840056534696670649481394206186/28934443)
(19030966014390119354747707312595/12925973563441, 9643749398956519809212686473306584070561587122/46472377899406087511)
(30109992590741616822755794395/21322132441, 256560670745641804607825115661199328845822/3113479101167261)
(150810791515265553135716001980/103064755369, -6243089641212891793045376462137503834195769/33087599869397653)
(2007345513017347986007170462239/1454612081329, -13550594453642125734812361156893769410736502/1754368356764711017)
(17360202434018790584827101980/12146905369, 157810275441064934092730411869752200905144/1338746881433597)
(16835872119900433822277227945895/6501434143681, 46645798122125846263268253749263043991510429442/16577298266650520671)
(9386174271786239019480976985795/5554501526401, -8786772290351399278282584205962045883148010302/13090843642920350399)

Commentary

Voznyy (2026), T = -3559/4025 for the rank-5 fibration of X163 given by y^2 + (9T^2 - 6T + 244)xy + (13T - 90)(22T - 15)(5T + 6)(4T + 15)y = x^3 mentioned in the curve#4645

last edited by Maksym Voznyy at 2026-10-25 16:42:16 UTC · history