curve #6273

y² = x³ + 433288x

a-invariants: [0, 0, 0, 433288, 0]
c-invariants: [-20797824, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 750953963776
discriminant (Δ): -5206069456905207808
Faltings height: 2.2808
naive height: 50.5511
primes of bad reduction: 2, 41, 1321
regulator: 3536.30384991839678997860919253951947545529877711
submitted by: The Wren
submitted at: 2026-10-31 23:34:55 UTC
last updated: 2026-10-31 23:34:55 UTC

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

(41, 4223)
(2952, 164328)
(6929, 579371)
(1321/36, 862613/216)
(5248/9, 573344/27)

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 23:34:56 UTC · history