curve #6271

y² = x³ − 264277x

a-invariants: [0, 0, 0, -264277, 0]
c-invariants: [12685296, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 2234954647328
discriminant (Δ): 1181294218663803712
Faltings height: 2.1572
naive height: 49.0679
primes of bad reduction: 2, 13, 29, 701
regulator: 1363.64861943101027371712925258058666874717003837
submitted by: The Wren
submitted at: 2026-10-31 23:32:48 UTC
last updated: 2026-10-31 23:32:48 UTC

Witness: 5 independent points log in to add more points →

(-468, 4602)
(-377, 6786)
(-116, 5394)
(-1856/9, 182584/27)
(3393/4, 157209/8)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 5. 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 23:32:48 UTC · history