curve #5828

y² = x³ − 10476720543126123

a-invariants: [0, 0, 0, 0, -10476720543126123]
c-invariants: [0, 9051886549260970272]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 41906882172504492
discriminant (Δ): -47417042882344719902013569524807728
Faltings height: 5.3326
naive height: 87.2990
primes of bad reduction: 2, 3, 11, 13, 17, 37, 73
regulator: 46301.4412093841881533087061170666320488513243934
submitted by: jesper-petersen
submitted at: 2026-10-25 11:02:37 UTC
last updated: 2026-10-25 11:02:37 UTC

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

(22156303, 104290736498)
(1720537/4, 2103022909/8)
(2352571/9, 2320126588/27)
(5934097/4, 14432250599/8)
(1430749411/529, 54104120829928/12167)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-10,17,-1,-7) and specialization t=29/16. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=39396786. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:02:37 UTC · history