curve #6313

y² = x³ + 6866558505990x

a-invariants: [0, 0, 0, 6866558505990, 0]
c-invariants: [-329594808287520, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 335286227315083526157369600
discriminant (Δ): -20720362465005151197268879975838195136000
Faltings height: 6.4255
naive height: 100.2867
primes of bad reduction: 2, 3, 5, 31, 173, 1733, 8209
regulator: 4981545650.0087368747120099407700589480242335071039927391
submitted by: The Wren
submitted at: 2026-11-01 00:01:03 UTC
last updated: 2026-11-01 00:01:03 UTC

Witness: 9 independent points log in to add more points →

(10, 8286470)
(6162250, 16622712550)
(8209/4, 949674583/8)
(1610361/4, 13457276739/8)
(53859249/100, 1963291308057/1000)
(213662785/1156, 44388400163345/39304)
(368502010/49, 7491035359970/343)
(44680675801/4356, 9747837429912179/287496)
(60847040718751/158785201, 3280153470782164768421/2000852317801)

Commentary

Quartic twist y^2 = x^3 + ax (j = 1728) of rank 9. 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 2026-11-01 00:01:03 UTC · history