curve #5801

y² = x³ − 196168949388

a-invariants: [0, 0, 0, 0, -196168949388]
c-invariants: [0, 169489972271232]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 65389649796
discriminant (Δ): -16624334896124416808203008
Faltings height: 3.5183
naive height: 65.5276
primes of bad reduction: 2, 3, 17, 23, 109
regulator: 1483.325421748638211339740304259151783052353908
submitted by: jesper-petersen
submitted at: 2026-10-25 10:55:25 UTC
last updated: 2026-10-25 10:55:25 UTC

Witness: 4 independent points log in to add more points →

(6213, 208953)
(10317, 949725)
(33681/4, 5064903/8)
(563857/4, 423387719/8)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-18,-1,19,-1) and specialization t=1/6. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=170476. PARI/GP 2.17.4 returned matching computational rank bounds [4,4].

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