curve #6309

y² = x³ − 3535090100242x

a-invariants: [0, 0, 0, -3535090100242, 0]
c-invariants: [169684324811616, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 799799169077055590941348096
discriminant (Δ): 2827362124786076755603056787831255839232
Faltings height: 6.2595
naive height: 98.2949
primes of bad reduction: 2, 41, 881, 4889, 10009
regulator: 2988567991.2726184555393340447102280804068892989213821980
submitted by: The Wren
submitted at: 2026-10-31 23:58:38 UTC
last updated: 2026-10-31 23:58:38 UTC

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

(-696200, 1457287020)
(-570888, 1353543732)
(-39147200800/177241, 65478291176733960/74618461)
(-1081065409/625, 15145745126889/15625)
(-551322752/361, 9295921836816/6859)
(-361535089/1225, 43250804232159/42875)
(-47161472/361, 4649970671184/6859)
(-156800/9, 6700344560/27)
(1953766809/484, 76419776544531/10648)

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 2026-10-31 23:58:38 UTC · history