curve #6289

y² = x³ − 333037562x

a-invariants: [0, 0, 0, -333037562, 0]
c-invariants: [15985802976, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 7098497132985846016
discriminant (Δ): 2364066179033595937676052992
Faltings height: 3.9420
naive height: 70.4849
primes of bad reduction: 2, 11, 13, 23, 197, 257
regulator: 107043.4625708874811292589528903323017676294209451164
submitted by: The Wren
submitted at: 2026-10-31 23:46:01 UTC
last updated: 2026-10-31 23:46:01 UTC

Witness: 7 independent points log in to add more points →

(-18216, 148764)
(-3289, 1029457)
(21528, 1675596)
(60138, 14052246)
(144992, 54770728)
(-36751/4, 12091079/8)
(167050/9, 12461930/27)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 7. 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 23:46:01 UTC · history