curve #6308

y² = x³ + 3482085701523x

a-invariants: [0, 0, 0, 3482085701523, 0]
c-invariants: [-167140113673104, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 387997466648029537424624928
discriminant (Δ): -2702080861684501454297700550300066730688
Faltings height: 6.2557
naive height: 98.2496
primes of bad reduction: 2, 3, 11, 13, 23, 349, 877, 1153
regulator: 767764487.82495193572714733898756450022317505887401680413
submitted by: The Wren
submitted at: 2026-10-31 23:58:03 UTC
last updated: 2026-10-31 23:58:03 UTC

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

(9238801/9, 58261273642/27)
(13937664/25, 181769351712/125)
(28420297/16, 219749175803/64)
(62773932/49, 878740887102/343)
(170404608/361, 9070605332976/6859)
(382448209/2025, 74275253211302/91125)
(931855753/1089, 68244121266274/35937)
(1006674097/144, 33058170671383/1728)
(6907377411/841, 588706566902982/24389)

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 23:58:03 UTC · history