curve #6298

y² = x³ + 26196991369x

a-invariants: [0, 0, 0, 26196991369, 0]
c-invariants: [-1257455585712, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 43922070834397471626304
discriminant (Δ): -1150626110557317192509708281370176
Faltings height: 5.0333
naive height: 83.5803
primes of bad reduction: 2, 41, 241, 617, 4297
regulator: 14985205.9917497732820801397587087979917882039089268228
submitted by: The Wren
submitted at: 2026-10-31 23:52:12 UTC
last updated: 2026-10-31 23:52:12 UTC

Witness: 8 independent points log in to add more points →

(496100, 367551470)
(7286153/16, 20872746587/64)
(9320193/64, 42540467583/512)
(24386308/729, 594989335738/19683)
(42458657/1024, 1114836956849/32768)
(169834628/22201, 46880895961178/3307949)
(219476772/529, 3490156207806/12167)
(3913243524/1225, 245110573576518/42875)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 8. Rank proven (independent generators plus a matching 2-descent upper bound) in M. Rubinstein's elliptic-curves-by-rank collection, file families/family_Z2.add_2026-10-24.part1.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by The Wren at 2026-10-31 23:52:12 UTC · history