curve #6286

y² = x³ − 253160002x

a-invariants: [0, 0, 0, -253160002, 0]
c-invariants: [12151680096, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 4101759143208960256
discriminant (Δ): 1038401352898298664826880512
Faltings height: 3.8734
naive height: 69.6622
primes of bad reduction: 2, 113, 593, 1889
regulator: 283725.5865569870415595024506814369109805936199770363
submitted by: The Wren
submitted at: 2026-10-31 23:43:48 UTC
last updated: 2026-10-31 23:43:48 UTC

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

(-15488, 453552)
(-4744, 1046052)
(-2825, 832245)
(58114, 13474146)
(-603081/121, 1419853701/1331)
(67009/4, 5427729/8)
(569873/16, 384884871/64)

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 23:43:48 UTC · history