curve #5805

y² = x³ − 58180781831481821763

a-invariants: [0, 0, 0, 0, -58180781831481821763]
c-invariants: [0, 50268195502400294003232]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 232723127325927287052
discriminant (Δ): -1462321457793713626406046890633073784969008
Faltings height: 6.7696
naive height: 104.5433
primes of bad reduction: 2, 3, 19, 149, 307, 563
regulator: 825038.423663842476395175662330830880449757985291
submitted by: jesper-petersen
submitted at: 2026-10-25 10:57:40 UTC
last updated: 2026-10-25 10:57:40 UTC

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

(3883243, 613902988)
(4542679, 5963357726)
(16458577/4, 27107334649/8)
(9398184553/1156, 860363252089813/39304)
(91397918464864393/9096581376, 26827333482518662379946587/867595545317376)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(18,-19,-1,1) and specialization t=9/13. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=2935877226. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:57:40 UTC · history