curve #5843

y² = x³ − 909861033675

a-invariants: [0, 0, 0, 0, -909861033675]
c-invariants: [0, 786119933095200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 294794974910700
discriminant (Δ): -357629947459260257090430000
Faltings height: 3.7740
naive height: 68.5963
primes of bad reduction: 2, 3, 5, 11, 17, 19, 31
regulator: 14320.4764422403334543800623053222977821648067847
submitted by: jesper-petersen
submitted at: 2026-10-25 11:06:02 UTC
last updated: 2026-10-25 11:06:02 UTC

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

(46531, 9991796)
(64201/4, 14366299/8)
(276241/16, 131730439/64)
(318649/16, 169197907/64)
(346921/4, 204193931/8)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-5,-16,-1,21) and specialization t=10/1. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=9912870. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:06:03 UTC · history