curve #5835

y² = x³ − 10912384282107

a-invariants: [0, 0, 0, 0, -10912384282107]
c-invariants: [0, 9428300019740448]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 3535612507402668
discriminant (Δ): -51442616471202391288475281968
Faltings height: 4.1881
naive height: 73.5650
primes of bad reduction: 2, 3, 7, 11, 17, 31, 47
regulator: 31455.9774658400561213811116046219126289334038867
submitted by: jesper-petersen
submitted at: 2026-10-25 11:04:11 UTC
last updated: 2026-10-25 11:04:11 UTC

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

(32767, 4926334)
(88753/4, 850423/8)
(390817/16, 122455921/64)
(531433/4, 386509333/8)
(718921/16, 571730117/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(23,-14,-9,-1) and specialization t=-2/19. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=34329834. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

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