curve #5813

y² = x³ − 2297681696568514347

a-invariants: [0, 0, 0, 0, -2297681696568514347]
c-invariants: [0, 1985196985835196395808]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 3063575595424685796
discriminant (Δ): -2280675389218257499640838829473769328688
Faltings height: 6.2310
naive height: 98.0800
primes of bad reduction: 2, 3, 7, 13, 89, 181, 199
regulator: 34884.0588496190135691480911594587490577730444833
submitted by: jesper-petersen
submitted at: 2026-10-25 10:59:27 UTC
last updated: 2026-10-25 10:59:27 UTC

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

(9833187, 30797541684)
(5294793/4, 1177713243/8)
(9547993/4, 26895784643/8)
(257427793/64, 4056744582233/512)
(9795258993/1936, 960809024860425/85184)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(17,-18,-1,1) and specialization t=11/6. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=583435762. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:59:28 UTC · history