curve #5803

y² + y = x³ − 13884038611

a-invariants: [0, 0, 1, 0, -13884038611]
c-invariants: [0, 11995809359688]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 18512051481
discriminant (Δ): -83275140158552213286723
Faltings height: 3.0769
naive height: 60.2312
primes of bad reduction: 3, 7, 11, 19, 31
regulator: 119.7943915878311603715459275448069081162756376
submitted by: jesper-petersen
submitted at: 2026-10-25 10:57:13 UTC
last updated: 2026-10-25 10:57:13 UTC

Witness: 4 independent points log in to add more points →

(2451, 28984)
(2821, 92550)
(12369/4, 1001885/8)
(4123, 237072)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-5,-3,-1,8) and specialization t=17/3. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=45353. PARI/GP 2.17.4 returned matching computational rank bounds [4,4].

last edited by jesper-petersen at 2026-10-25 10:57:13 UTC · history