curve #5807

y² = x³ − 303287677984124427

a-invariants: [0, 0, 0, 0, -303287677984124427]
c-invariants: [0, 262040553778283504928]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 98265207666856314348
discriminant (Δ): -39736835546544843547423555793409838128
Faltings height: 5.8935
naive height: 94.0301
primes of bad reduction: 2, 3, 11, 43, 67, 79, 127
regulator: 295713.526107895359815021481692870489476144241806
submitted by: jesper-petersen
submitted at: 2026-10-25 10:58:06 UTC
last updated: 2026-10-25 10:58:06 UTC

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

(6313627/9, 5529589660/27)
(67022233/36, 535641971165/216)
(401651089/64, 8044643776441/512)
(932356657/256, 28379552556631/4096)
(3204430629607/32041, 5736225893801497826/5735339)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(-23,3,-1,20) and specialization t=1/20. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=5723204454. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 10:58:06 UTC · history