curve #6272

y² = x³ − 319945x

a-invariants: [0, 0, 0, -319945, 0]
c-invariants: [15357360, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3275673696800
discriminant (Δ): 2096070841845352000
Faltings height: 2.2050
naive height: 49.6413
primes of bad reduction: 2, 5, 61, 1049
regulator: 2024.62092976310497185238302313778716716312557525
submitted by: The Wren
submitted at: 2026-10-31 23:34:15 UTC
last updated: 2026-10-31 23:34:15 UTC

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

(-245, 7980)
(-61, 4392)
(605, 5280)
(-1220/9, 172630/27)
(14945/16, 1453935/64)

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 23:34:15 UTC · history