curve #6306

y² = x³ + 585688506100x

a-invariants: [0, 0, 0, 585688506100, 0]
c-invariants: [-28113048292800, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 27442482094211978976800
discriminant (Δ): -12858197073148010738917278846784000000
Faltings height: 5.8101
naive height: 92.9018
primes of bad reduction: 2, 5, 101, 181, 421, 761
regulator: 58407026.1827057464919211563579603763746407485242283471
submitted by: The Wren
submitted at: 2026-10-31 23:56:54 UTC
last updated: 2026-10-31 23:56:54 UTC

Witness: 8 independent points log in to add more points →

(412580, 558456500)
(1472580, 2013889500)
(2203856, 3463359704)
(1640745/4, 4448865675/8)
(12480545/16, 61768162225/64)
(46101605/49729, 258405300636175/11089567)
(2588678480/44521, 1738549995471800/9393931)
(5788565525/49, 440418490064825/343)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 8. 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 23:56:54 UTC · history