curve #5816

y² + y = x³ − 752834503587993577

a-invariants: [0, 0, 1, 0, -752834503587993577]
c-invariants: [0, 650449011100026450312]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1003779338117324769
discriminant (Δ): -244840229190394868268846859737279516723
Faltings height: 6.0450
naive height: 95.8484
primes of bad reduction: 3, 7, 11, 13, 37, 71, 127
regulator: 20595.2038144225541773897094243997544345904727118
submitted by: jesper-petersen
submitted at: 2026-10-25 11:00:09 UTC
last updated: 2026-10-25 11:00:09 UTC

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

(5164201/4, 9462691129/8)
(1707069, 2054679763)
(10163937/4, 31651387925/8)
(8979789, 26895109468)
(70687057/4, 594265101405/8)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(20,-4,-1,-16) and specialization t=25/3. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=333962629. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:00:09 UTC · history