curve #5831

y² + y = x³ − 68874511159477

a-invariants: [0, 0, 1, 0, -68874511159477]
c-invariants: [0, 59507577641787912]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 22315341615670467
discriminant (Δ): -2049277660181375343698016197523
Faltings height: 4.4951
naive height: 77.2498
primes of bad reduction: 3, 7, 13, 31, 43, 79
regulator: 1755.65697467045137442309409905097152493433201033
submitted by: jesper-petersen
submitted at: 2026-10-25 11:03:19 UTC
last updated: 2026-10-25 11:03:19 UTC

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

(121303, 41424974)
(166453, 67401575)
(737149, 632842416)
(4234321/64, 7606845991/512)
(1368991/25, 1220455414/125)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-9,-1,7,2) and specialization t=13/2. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=86246433. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:03:19 UTC · history