curve #5814

y² = x³ − 4599366742683075

a-invariants: [0, 0, 0, 0, -4599366742683075]
c-invariants: [0, 3973852865678176800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1490194824629316300
discriminant (Δ): -9138603355358019597669553108830000
Faltings height: 5.1954
naive height: 85.6525
primes of bad reduction: 2, 3, 5, 11, 19, 89, 421
regulator: 173731.117455448734639200737065552263677226761062
submitted by: jesper-petersen
submitted at: 2026-10-25 10:59:42 UTC
last updated: 2026-10-25 10:59:42 UTC

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

(417211, 260811184)
(2875009/4, 4844536673/8)
(3733849/16, 5763412957/64)
(11513929/64, 17908423483/512)
(12868441/36, 43776526339/216)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-20,-3,-1,23) and specialization t=-13/7. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=704791890. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:59:42 UTC · history