curve #6269

y² = x³ − 186386x

a-invariants: [0, 0, 0, -186386, 0]
c-invariants: [8946528, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 2223343423744
discriminant (Δ): 414400087377949184
Faltings height: 2.0699
naive height: 48.0203
primes of bad reduction: 2, 41, 2273
regulator: 2966.18685790221646762225682133333278864308078959
submitted by: The Wren
submitted at: 2026-10-31 23:30:41 UTC
last updated: 2026-10-31 23:30:41 UTC

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

(-369, 4305)
(-328, 5084)
(-25, 2155)
(-20992/49, 379168/343)
(2601/4, 99195/8)

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 23:30:41 UTC · history