curve #5844

y² = x³ − 2226495060300

a-invariants: [0, 0, 0, 0, -2226495060300]
c-invariants: [0, 1923691732099200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 180346099884300
discriminant (Δ): -2141545069529409874790880000
Faltings height: 3.9232
naive height: 70.3860
primes of bad reduction: 2, 3, 5, 7, 31, 397
regulator: 6145.51980191549635433833323870312202217913021595
submitted by: jesper-petersen
submitted at: 2026-10-25 11:06:15 UTC
last updated: 2026-10-25 11:06:15 UTC

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

(28861, 4670491)
(86149, 25241657)
(52801/4, 2170399/8)
(529201/36, 210535849/216)
(1053241/64, 764666861/512)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-5,-1,1,4) and specialization t=-5/4. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=15506820. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:06:15 UTC · history