curve #5842

y² = x³ − 2099773974675

a-invariants: [0, 0, 0, 0, -2099773974675]
c-invariants: [0, 1814204714119200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 680326767794700
discriminant (Δ): -1904709921720097337865630000
Faltings height: 3.9134
naive height: 70.2688
primes of bad reduction: 2, 3, 5, 13, 61, 211
regulator: 6941.57329410160688689807818460799819445733095574
submitted by: jesper-petersen
submitted at: 2026-10-25 11:05:47 UTC
last updated: 2026-10-25 11:05:47 UTC

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

(12871, 180194)
(100711, 31927766)
(6190951, 15404090026)
(2561329/196, 996511433/2744)
(7194889/484, 11592304763/10648)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-14,-1,1,13) and specialization t=-1/14. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=15059070. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:05:48 UTC · history