curve #6318

y² = x³ − 27466234023960x

a-invariants: [0, 0, 0, -27466234023960, 0]
c-invariants: [1318379233150080, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 335286227315083526157369600
discriminant (Δ): 1326103197760329676625208318453644488704000
Faltings height: 6.7720
naive height: 104.4455
primes of bad reduction: 2, 3, 5, 31, 173, 1733, 8209
regulator: 1270802461.7369226721204106991760354459245493640571410049
submitted by: The Wren
submitted at: 2026-11-01 00:04:26 UTC
last updated: 2026-11-01 00:04:26 UTC

Witness: 9 independent points log in to add more points →

(-3475224, 7313029704)
(223057896, 3330477445176)
(541233960, 12590912243880)
(-686655850599/170569, 474287615781591879/70444997)
(-9612951/16, 258270707403/64)
(767511040/9, 21222927461440/27)
(5576669224/9, 416434587634072/27)
(69867463929/7921, 14854403035027593/704969)
(18174680139731944/1577440089, 2182040102961846219543688/62651188014813)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 9. 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 2026-11-01 00:04:27 UTC · history