curve #5829

y² = x³ − 29044961271618075

a-invariants: [0, 0, 0, 0, -29044961271618075]
c-invariants: [0, 25094846538678016800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 38726615028824100
discriminant (Δ): -364439422916550949384740384016830000
Faltings height: 5.5025
naive height: 89.3384
primes of bad reduction: 2, 3, 5, 7, 19, 31, 37, 43
regulator: 37616.9980416738209327876586690134560733213161393
submitted by: jesper-petersen
submitted at: 2026-10-25 11:02:52 UTC
last updated: 2026-10-25 11:02:52 UTC

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

(746179, 621622792)
(5495959/9, 12034731898/27)
(45917851/121, 212980112324/1331)
(396950071/9, 7908675427394/27)
(109592790951/196249, 33117115324935474/86938307)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-6,-1,11,-5) and specialization t=-29/10. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=65596930. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:02:52 UTC · history