curve #5815

y² + y = x³ − 1055324902331913019

a-invariants: [0, 0, 1, 0, -1055324902331913019]
c-invariants: [0, 911800715614772848200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 1407099869775884025
discriminant (Δ): -481123000576164276898517697270798751875
Faltings height: 6.1013
naive height: 96.5239
primes of bad reduction: 3, 5, 7, 43, 59, 61, 73
regulator: 173998.399982224274754224626480824012654270730044
submitted by: jesper-petersen
submitted at: 2026-10-25 10:59:55 UTC
last updated: 2026-10-25 10:59:55 UTC

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

(1211581, 850405433)
(36847699/25, 183140982168/125)
(89803651/25, 841278591976/125)
(3467493211/2809, 135280575655683/148877)
(13491784299/13225, 121881794481157/1520875)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-8,-6,14,-1) and specialization t=31/9. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=395404135. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:59:55 UTC · history