curve #6312

y² = x³ + 4429942099363x

a-invariants: [0, 0, 0, 4429942099363, 0]
c-invariants: [-212637220769424, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 627980384118677240480184608
discriminant (Δ): -5563833482362952398462793217130638409408
Faltings height: 6.3159
naive height: 98.9718
primes of bad reduction: 2, 11, 97, 881, 1697, 2777
regulator: 799642075.27976482536796178669527068763967341377651096635
submitted by: The Wren
submitted at: 2026-11-01 00:00:26 UTC
last updated: 2026-11-01 00:00:26 UTC

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

(709676, 1871161358)
(7064739, 19593399474)
(9346529, 29289844054)
(45523361, 307478787382)
(67222067, 551417527078)
(237314089/9, 3667451931406/27)
(2135826801/10609, 1036654863325098/1092727)
(2847499699/3249, 395277599668094/185193)
(17552699278796/2550409, 76900704697976648918/4073003173)

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 00:00:26 UTC · history