curve #5868

y² = x³ − 2413186503763710675

a-invariants: [0, 0, 0, 0, -2413186503763710675]
c-invariants: [0, 2084993139251846023200]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 781872427219442258700
discriminant (Δ): -2515738652041242929384430595487068830000
Faltings height: 6.2392
naive height: 98.1781
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 83, 127
regulator: 11646849.789127882394709675606628440880700035108843
submitted by: jesper-petersen
submitted at: 2026-10-25 17:26:13 UTC
last updated: 2026-10-25 17:26:13 UTC

Witness: 6 independent points log in to add more points →

(911997529/324, 26008970036867/5832)
(10374125371/3249, 1016722847052856/185193)
(97279711/9, -958557238066/27)
(200699041/144, -937252456561/1728)
(5613958879/2025, 396098463017258/91125)
(360228619/81, 6742586838772/729)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, on the plane b=-1, c=-a-d. Specialization (a,b,c,d)=(20,-1,-18,-2), t=-76/129, w=16676660/16641 on w²=D_{a,d}(t), where k=(a³+1)(t-a)w/8 up to rational cubic scaling, gives the cubesum curve u³+v³=16143857730. PARI/GP 2.17.4 ellrank bounds [6,6].

last edited by jesper-petersen at 2026-10-25 17:26:13 UTC · history