curve #6288

y² = x³ − 330357986x

a-invariants: [0, 0, 0, -330357986, 0]
c-invariants: [15857183328, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 6984729530494476544
discriminant (Δ): 2307461180448880855200080384
Faltings height: 3.9400
naive height: 70.4607
primes of bad reduction: 2, 7, 113, 367, 569
regulator: 196415.8409043041072332830452136312195951911880878322
submitted by: The Wren
submitted at: 2026-10-31 23:45:19 UTC
last updated: 2026-10-31 23:45:19 UTC

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

(-6606, 1376250)
(-2569, 911995)
(18306, 294930)
(19775, 1095535)
(56952, 12880644)
(-373239/256, 2833506075/4096)
(-82208/9, 40549096/27)

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 23:45:19 UTC · history