curve #5830

y² = x³ − 7185178467087075

a-invariants: [0, 0, 0, 0, -7185178467087075]
c-invariants: [0, 6207994195563232800]
rank (lower bound): ≥ 5
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 28740713868348300
discriminant (Δ): -22302773108881244173023393048030000
Faltings height: 5.2697
naive height: 86.5447
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 37
regulator: 21238.3635736601577062395256799839219396225474822
submitted by: jesper-petersen
submitted at: 2026-10-25 11:03:05 UTC
last updated: 2026-10-25 11:03:05 UTC

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

(2750839, 4561658738)
(831649/4, 339632657/8)
(4286191/9, 8573533514/27)
(6589219/9, 16758622972/27)
(10043761/16, 31364874359/64)

Commentary

Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with (a,b,c,d)=(17,-10,-7,-1) and specialization t=23/7. The square-condition construction yields, via 3-isogeny and rational cubic scaling, the cubesum curve u^3+v^3=32626230. PARI/GP 2.17.4 returned matching computational rank bounds [5,5].

last edited by jesper-petersen at 2026-10-25 11:03:05 UTC · history