curve #5839

y² + y = x³ − 3030331280419

a-invariants: [0, 0, 1, 0, -3030331280419]
c-invariants: [0, 2618206226281800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 981827334855675
discriminant (Δ): -3967016113044435245979751875
Faltings height: 3.9745
naive height: 71.0025
primes of bad reduction: 3, 5, 7, 11, 23, 227
regulator: 63068.1077428994958696301523354620948252863071857
submitted by: jesper-petersen
submitted at: 2026-10-25 11:05:06 UTC
last updated: 2026-10-25 11:05:06 UTC

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

(18781, 1895841)
(38899, 7471880)
(56329, 13255154)
(328909, 188623214)
(480781/25, 252554446/125)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-4,0,-1,4) and specialization t=-29/6. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=18090765. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:05:06 UTC · history