curve #4748

y² + y = x³ − 1192930094217270275562649590693370774256731894205269

a-invariants: [0, 0, 1, 0, -1192930094217270275562649590693370774256731894205269]
c-invariants: [0, 1030691601403721518086129246359072348957816356593352200]
rank (lower bound): ≥ 11
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 4771720376869081102250598362773483097026927576821075 ☆ record for cube-sum curves, rank ≥ 11
discriminant (Δ): -614771514585745345388702014937083440173787277052610313846784425214738511773064284865564132181269232201875 ☆ record for cube-sum curves, rank ≥ 11
Faltings height: 18.7860 ☆ record for cube-sum curves, rank ≥ 11
naive height: 248.7397 ☆ record for cube-sum curves, rank ≥ 11
primes of bad reduction: 3, 5, 7, 13, 19, 23, 31, 43, 59, 61, 73, 79, 103, 109, 157, 457
regulator: 71633425232431364.3461603726033451688129675673690537516273911
submitted by: David Renshaw
submitted at: 2026-10-20 21:28:04 UTC
last updated: 2026-10-20 21:28:04 UTC

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

(23642591593617729121/144, 98252123534609206746112660367/1728)
(88310759603519640769/64, -829701375518392147007908984353/512)
(186947444570606201329/484, 2529513158175975931208754740643/10648)
(367546341152848996609/324, 7043534341799824998039842493011/5832)
(6021515587456251202009/676, 467259732332479638700440654732135/17576)
(8127471096051103790569/64516, 465319221889540295783669347447015/16387064)
(53988119916600845996329/41616, -12540897742116496657715664117492149/8489664)
(66647515276479419963521/592900, 6885324121542825089712908509944819/456533000)
(119330278066293515680489/692224, 36104560562310529545792372775365003/575930368)
(206456077678646367560641/1156, -93808352391299389272941929514640813/39304)
(610310937232689051741769/3168400, -435183850967477439582792610957898853/5639752000)

Commentary

Cube-sum curve x^3 + y^3 = 13293998056584952174157235 (Weierstrass model y^2 = x^3 - 432*13293998056584952174157235^2). Source: Elkies & Rogers, 'Elliptic Curves x^3+y^3=k of High Rank', ANTS-VI (LNCS 3076, 2004), arXiv:math/0403116, https://arxiv.org/abs/math/0403116 Points mapped from the 3-isogenous curve uv(u+v)=A given there via the dual isogeny.

last edited by David Renshaw at 2026-10-20 21:28:04 UTC · history