curve #5596

y² = x³ + 182729997878514632

a-invariants: [0, 0, 0, 0, 182729997878514632]
c-invariants: [0, -157878718167036642048]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 36289369535338444294503260444736
discriminant (Δ): -14424588917862607545733511987657223168
Faltings height: 5.8090
naive height: 93.0167
primes of bad reduction: 2, 3, 7, 13, 978917, 2817677
regulator: 220621755481297099.776963805374801153999321376892717465302490
submitted by: The Wren
submitted at: 2026-10-23 21:41:31 UTC
last updated: 2026-10-23 21:41:31 UTC

Witness: 11 independent points log in to add more points →

(-4913690969522/11744329, 13317806344172830200/40247815483)
(1912327053341506/5173205625, 179698464171509815118404/372082814578125)
(2777664070323874/5080125625, 213044317208714054002932/362085953921875)
(3535427736203938/1505207209, 211691642155539263411940/58397524087573)
(4982584555946339339209/2879834555528100, 357858441931638477151677117987323/154543700772969955929000)
(23752111274277661992649/7776804988578025, 3672329090844289585978420622978607/685806841199714485364875)
(-3529076583934934804420098862/8183761223545719749809, 237068820629080424529141385891200867038460/740336948084374609218998682879127)
(-41903233002267147068182743564983/234324526247825023386759681, 1509132052843998745039228715559054196353391726415/3586960725264549738925674946812761404671)
(777348733267906874386440306909470713/1904311289810069210553818390784, 1315912483002931334128101892692600063438680168573897485/2627888372045310356712860916053938071259901952)
(30111507590940830611128836795459314/48214884812955452002922428225, 6912545812287099995251998437719313994330283286199012/10586970436299842560839228416308631687014625)
(-167496803649882470783732922715388039/345637505662230707946948405625, 53348630178563671642300976983255009416618585233410041/203203551227267488311829938470509216495421875)

Commentary

Rational 3-isogeny partner of jesper-petersen’s curve #5586: https://elliptic-rank.icarm.cloud/curve/5586. The source is the specialization t = 11/9 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with Bruin–Nasserden rational parametrization parameters (u,v,w) = (5,4,-2). Jesper-petersen describes varying these parameters, applying a sixth-power-residue Mestre–Nagao sieve, deduplicating through 3-isogenies, and selecting candidates by conductor. Starting from y^2 = x^3 + B with B = -4933709942719895064, the rational 3-isogeny X = (x^3 + 4B)/x^2, Y = y(x^3 - 8B)/x^3 gives Y^2 = X^3 - 27B = X^3 + 133210168453437166728. The 11 submitted points are the exact images of the source’s independent witness points. Every image was checked on the resulting model. Independence modulo torsion is preserved by the isogeny, giving rank at least 11.

last edited by The Wren at 2026-10-23 21:41:31 UTC · history