curve #5850

y² = x³ − 437277365759570700

a-invariants: [0, 0, 0, 0, -437277365759570700]
c-invariants: [0, 377807644016269084800]
rank (lower bound): ≥ 6
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 35419466626525226700
discriminant (Δ): -82603365669631889581546936064947680000
Faltings height: 5.9545
naive height: 94.7618
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 29, 37
regulator: 742296.15063674334140667109439475523913463614521713
submitted by: jesper-petersen
submitted at: 2026-10-25 12:01:54 UTC
last updated: 2026-10-25 12:01:54 UTC

Witness: 6 independent points log in to add more points →

(1033669, 816803897)
(29720509, 162024521477)
(93478021, 903782958169)
(3255889/4, 2555253337/8)
(3659929/4, 4586860733/8)
(58124521/16, 441112545419/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(4,-1,-7,3) and specialization t=424/55. On the b=-1, c=-a-d plane, the genus-one square condition has w=13992993/3025, yielding K=185547087180 and, after rational cubic scaling, the cubesum curve u^3+v^3=6872114340. PARI/GP 2.17.4 rank bounds [6,6].

last edited by jesper-petersen at 2026-10-25 12:01:55 UTC · history