curve #5808

y² = x³ − 125310163377924430668

a-invariants: [0, 0, 0, 0, -125310163377924430668]
c-invariants: [0, 108267981158526708097152]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 41770054459308143556
discriminant (Δ): -6783539203786512885208747083532197616128768
Faltings height: 6.8975
naive height: 106.0778
primes of bad reduction: 2, 3, 7, 19, 23, 31, 37, 307
regulator: 26349.6998200195361254497082683028414896880109864
submitted by: jesper-petersen
submitted at: 2026-10-25 10:58:21 UTC
last updated: 2026-10-25 10:58:21 UTC

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

(7394709, 16704647631)
(20071353/4, 8128897965/8)
(33930561/4, 176192583777/8)
(327913761/64, 1552552105167/512)
(1147218323421/21025, 1228291791394074219/3048625)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-5,-13,-1,18) and specialization t=1/18. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=4308650444. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:58:21 UTC · history