curve #5802

y² = x³ − 20640614427

a-invariants: [0, 0, 0, 0, -20640614427]
c-invariants: [0, 17833490864928]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 27520819236
discriminant (Δ): -184047104415202792558128
Faltings height: 3.1430
naive height: 61.0242
primes of bad reduction: 2, 3, 43, 643
regulator: 369.8051155229239477987213319928894934377045334
submitted by: jesper-petersen
submitted at: 2026-10-25 10:55:39 UTC
last updated: 2026-10-25 10:55:39 UTC

Witness: 4 independent points log in to add more points →

(13503, 1562490)
(19651, 2750968)
(82947, 23888736)
(857119/289, 362588986/4913)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(7,-12,5,-1) and specialization t=1/3. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=55298. PARI/GP 2.17.4 returned matching computational rank bounds [4,4].

last edited by jesper-petersen at 2026-10-25 10:55:39 UTC · history