curve #5845

y² = x³ − 59789521683675

a-invariants: [0, 0, 0, 0, -59789521683675]
c-invariants: [0, 51658146734695200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 79719362244900
discriminant (Δ): -1544307942166261778920970430000
Faltings height: 4.4716
naive height: 76.9668
primes of bad reduction: 2, 3, 5, 7, 17, 41, 61
regulator: 14610.0374615499875128624393523958649436059135304
submitted by: jesper-petersen
submitted at: 2026-10-25 11:06:27 UTC
last updated: 2026-10-25 11:06:27 UTC

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

(39711, 1683234)
(67039, 15540238)
(92659, 27124748)
(1407819/25, 1362358872/125)
(1412089/16, 1603371763/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(14,-18,-1,4) and specialization t=7/26. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=2976190. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:06:27 UTC · history