curve #5833

y² = x³ − 4606084686048300

a-invariants: [0, 0, 0, 0, -4606084686048300]
c-invariants: [0, 3979657168745731200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 4606084686048300
discriminant (Δ): -9165318970341023874721411008480000
Faltings height: 5.1956
naive height: 85.6555
primes of bad reduction: 2, 3, 5, 7, 13, 31, 463
regulator: 396072.454155119951507171017505405621457504379434
submitted by: jesper-petersen
submitted at: 2026-10-25 11:03:45 UTC
last updated: 2026-10-25 11:03:45 UTC

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

(15171121/36, 57244457431/216)
(9811093621/1521, 971789949540431/59319)
(21165807481/47524, 2997950715166771/10360232)
(22689441229/43681, 3361082503752883/9129329)
(220996352941/958441, 82085194642698611/938313739)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-21,-1,6,15) and specialization t=-12/7. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=26122460. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:03:46 UTC · history