curve #5837

y² = x³ − 9871298226075

a-invariants: [0, 0, 0, 0, -9871298226075]
c-invariants: [0, 8528801667328800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 3198300625248300
discriminant (Δ): -42095172384624142861879230000
Faltings height: 4.1714
naive height: 73.3645
primes of bad reduction: 2, 3, 5, 11, 13, 43, 59
regulator: 34856.2883180727808876607744757538755308876866187
submitted by: jesper-petersen
submitted at: 2026-10-25 11:04:39 UTC
last updated: 2026-10-25 11:04:39 UTC

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

(27391, 3267914)
(111241/4, 27290939/8)
(899431/25, 757218046/125)
(3062761/64, 5112977909/512)
(12118561/100, 42069641809/1000)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-6,23,-1,-17) and specialization t=30/7. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=32651190. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:04:39 UTC · history