curve #5841

y² = x³ − 2207467666803

a-invariants: [0, 0, 0, 0, -2207467666803]
c-invariants: [0, 1907252064117792]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 715219524044172
discriminant (Δ): -2105098631991654031400029488
Faltings height: 3.9217
naive height: 70.3689
primes of bad reduction: 2, 3, 7, 31, 59, 67
regulator: 8057.09809313981616579530992103564851871704113386
submitted by: jesper-petersen
submitted at: 2026-10-25 11:05:34 UTC
last updated: 2026-10-25 11:05:34 UTC

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

(14539, 930496)
(34699, 6290536)
(39463, 7697362)
(2569249/196, 581815471/2744)
(3503899/225, 4227766732/3375)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(5,-2,-1,-3) and specialization t=15/13. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=15440418. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:05:34 UTC · history