curve #6268

y² = x³ − 108322x

a-invariants: [0, 0, 0, -108322, 0]
c-invariants: [5199456, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 750953963776
discriminant (Δ): 81344835264143872
Faltings height: 1.9343
naive height: 46.3922
primes of bad reduction: 2, 41, 1321
regulator: 1768.15192495919839498930459626975973772764938856
submitted by: The Wren
submitted at: 2026-10-31 23:29:25 UTC
last updated: 2026-10-31 23:29:25 UTC

Witness: 5 independent points log in to add more points →

(-328, 492)
(-41, 2091)
(-9, 987)
(1058, 32706)
(2009/4, 68019/8)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 5. 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 23:29:25 UTC · history