curve #5825

y² + y = x³ − 54280335565725019

a-invariants: [0, 0, 1, 0, -54280335565725019]
c-invariants: [0, 46898209928786416200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 72373780754300025
discriminant (Δ): -1272825286183171753044779234745151875
Faltings height: 5.6067
naive height: 90.5890
primes of bad reduction: 3, 5, 7, 11, 13, 19, 23, 41
regulator: 28700.2637895375604292009041117389572487791251765
submitted by: jesper-petersen
submitted at: 2026-10-25 11:01:56 UTC
last updated: 2026-10-25 11:01:56 UTC

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

(378651, 3057736)
(658749, 481231705)
(1246609, 1372221714)
(11707059/25, 27502425752/125)
(49433839/121, 156973774884/1331)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(10,-4,-1,-6) and specialization t=-16/1. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=89674585. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:01:56 UTC · history