curve #5811

y² = x³ − 21567098772201675

a-invariants: [0, 0, 0, 0, -21567098772201675]
c-invariants: [0, 18633973339182247200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 6987740002193342700
discriminant (Δ): -200940371762358094822250009052030000
Faltings height: 5.4529
naive height: 88.7430
primes of bad reduction: 2, 3, 5, 7, 11, 19, 67, 173
regulator: 70465.4767133989601240767279464660103536732038454
submitted by: jesper-petersen
submitted at: 2026-10-25 10:59:02 UTC
last updated: 2026-10-25 10:59:02 UTC

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

(360259, 158712548)
(1842031, 2495713954)
(2873161/4, 4726288741/8)
(12428299/25, 39783434468/125)
(25634401/64, 105788735249/512)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-7,-2,9,-1) and specialization t=-17/24. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=1526186970. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:59:02 UTC · history