curve #5827

y² + y = x³ − 14429373306936919

a-invariants: [0, 0, 1, 0, -14429373306936919]
c-invariants: [0, 12466978537193497800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 57717493227747675
discriminant (Δ): -89945343661367665629716943633451875
Faltings height: 5.3859
naive height: 87.9392
primes of bad reduction: 3, 5, 7, 31, 43, 991
regulator: 3828.37070235659603165721069402620592821597887674
submitted by: jesper-petersen
submitted at: 2026-10-25 11:02:23 UTC
last updated: 2026-10-25 11:02:23 UTC

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

(306219, 119519059)
(583699, 429464219)
(3963009, 7888369414)
(3963009/16, 1771464991/64)
(6174921/16, 13279490149/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-6,-1,0,6) and specialization t=-4/11. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=46235105. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:02:23 UTC · history