curve #5798

y² = x³ − 2046828

a-invariants: [0, 0, 0, 0, -2046828]
c-invariants: [0, 1768459392]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 165793068
discriminant (Δ): -1809866100204288
Faltings height: 1.6066
naive height: 42.5867
primes of bad reduction: 2, 3, 7, 59
regulator: 138.06675967163682156893120853026323398116052
submitted by: jesper-petersen
submitted at: 2026-10-25 10:54:45 UTC
last updated: 2026-10-25 10:54:45 UTC

Witness: 3 independent points log in to add more points →

(133, 553)
(973, 30317)
(3409/16, 176729/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(2,-17,-1,15) and specialization t=-1/13. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=14868. PARI/GP 2.17.4 returned matching computational rank bounds [3,3].

last edited by jesper-petersen at 2026-10-25 10:54:45 UTC · history