curve #5818

y² = x³ − 795331225200368268

a-invariants: [0, 0, 0, 0, -795331225200368268]
c-invariants: [0, 687166178573118183552]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 265110408400122756
discriminant (Δ): -273262359360406567011530706498810163968
Faltings height: 6.0542
naive height: 95.9582
primes of bad reduction: 2, 3, 31, 71, 127, 307
regulator: 38459.0697776859204189147423142260644981324610652
submitted by: jesper-petersen
submitted at: 2026-10-25 11:00:36 UTC
last updated: 2026-10-25 11:00:36 UTC

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

(1846453, 2345196097)
(2222373, 3190742793)
(8460613/9, 5082411095/27)
(556645953/16, 13133016277857/64)
(309415495341/126025, 167424299771191389/44738875)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-17,-1,20,-3) and specialization t=3/4. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=343259156. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:00:36 UTC · history