curve #5809

y² = x³ − 8097129730615635675

a-invariants: [0, 0, 0, 0, -8097129730615635675]
c-invariants: [0, 6995920087251909223200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 32388518922462542700
discriminant (Δ): -28323436265749283077797656882831728830000
Faltings height: 6.4409
naive height: 100.5992
primes of bad reduction: 2, 3, 5, 7, 17, 19, 67, 241
regulator: 115644.674687303589424606715110551788100162826912
submitted by: jesper-petersen
submitted at: 2026-10-25 10:58:33 UTC
last updated: 2026-10-25 10:58:33 UTC

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

(2147551, 1344366926)
(62839081/16, 463648556629/64)
(1200481009/16, 41593817161673/64)
(1458187129/256, 54449151939917/4096)
(4383151591/729, 284731790870114/19683)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-11,-5,-1,16) and specialization t=1/16. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=1095251010. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:58:34 UTC · history