curve #5834

y² + y = x³ − 11482074432397

a-invariants: [0, 0, 1, 0, -11482074432397]
c-invariants: [0, 9920512309590792]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 3720192116096547
discriminant (Δ): -56954030373114832232330145363
Faltings height: 4.1966
naive height: 73.6668
primes of bad reduction: 3, 7, 13, 19, 31, 73
regulator: 1260.74832695017990891227949818346262368382215100
submitted by: jesper-petersen
submitted at: 2026-10-25 11:04:00 UTC
last updated: 2026-10-25 11:04:00 UTC

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

(126217/4, 35719407/8)
(145483, 55386909)
(205933, 93390615)
(300979, 165086981)
(3912727/121, 6289707987/1331)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-8,-7,-1,15) and specialization t=7/1. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=35214543. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:04:00 UTC · history