curve #5826

y² + y = x³ − 49272667248196519

a-invariants: [0, 0, 1, 0, -49272667248196519]
c-invariants: [0, 42571584502441792200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 65696889664262025
discriminant (Δ): -1048807758708647092565650549275451875
Faltings height: 5.5906
naive height: 90.3954
primes of bad reduction: 3, 5, 13, 31, 109, 389
regulator: 25512.1120423346819417345305703323115645258132787
submitted by: jesper-petersen
submitted at: 2026-10-25 11:02:10 UTC
last updated: 2026-10-25 11:02:10 UTC

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

(519051, 300943363)
(2199729/4, 2736898763/8)
(571051, 370063011)
(25068801/64, 53271222143/512)
(109246449/25, 1141517924582/125)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-5,12,-1,-7) and specialization t=29/15. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=85438015. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:02:10 UTC · history