curve #5812

y² = x³ − 13791632620440675

a-invariants: [0, 0, 0, 0, -13791632620440675]
c-invariants: [0, 11915970584060743200]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 4468488969022778700
discriminant (Δ): -82170344305671834157847076004830000
Faltings height: 5.3784
naive height: 87.8488
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 31
regulator: 42936.9669693206778357654217852393336868081854126
submitted by: jesper-petersen
submitted at: 2026-10-25 10:59:14 UTC
last updated: 2026-10-25 10:59:14 UTC

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

(240331, 9468004)
(261079, 63277942)
(1039129/4, 489260317/8)
(4441801/4, 9314090149/8)
(43157569/4, 283519592447/8)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-16,-5,-1,21) and specialization t=10/1. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=1220449230. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:59:14 UTC · history