curve #5838

y² = x³ − 5192086447443

a-invariants: [0, 0, 0, 0, -5192086447443]
c-invariants: [0, 4485962690590752]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1682236008971532
discriminant (Δ): -11645753044775589677334923568
Faltings height: 4.0643
naive height: 72.0795
primes of bad reduction: 2, 3, 7, 71, 2647
regulator: 35079.3986695157083859852498003155567691059404910
submitted by: jesper-petersen
submitted at: 2026-10-25 11:04:52 UTC
last updated: 2026-10-25 11:04:52 UTC

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

(51499, 11462584)
(129703, 46656022)
(1191463, 1300529398)
(79177/4, 12808859/8)
(6625441/100, 16900928239/1000)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(5,-3,-1,-2) and specialization t=16/9. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=23680062. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:04:53 UTC · history