curve #6218

y² = x³ + 396021808630297601

a-invariants: [0, 0, 0, 0, 396021808630297601]
c-invariants: [0, -342162842656577127264]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 5645997824789233996179850145748787236
discriminant (Δ): -67751973897470807954158201748985446832
Faltings height: 5.9380
naive height: 94.5636
primes of bad reduction: 2, 3, 160481, 2467717727521
regulator: 12490181499378.381221244063242044254337466056404116047750160205
submitted by: jesper-petersen
submitted at: 2026-10-30 22:12:12 UTC
last updated: 2026-10-30 22:12:12 UTC

Witness: 12 independent points log in to add more points →

(-185548, 624206497)
(3208040, 5780279551)
(-24159480745/52441, 6558291741275564/12008989)
(7018106/169, 1382702895985/2197)
(66167758/289, 3138263464059/4913)
(123861455/2401, 74049657875276/117649)
(425046340/169, 8871429414453/2197)
(902652598/169, 27154674576951/2197)
(7820921111/6889, 779651390254720/571787)
(39539229935/9409, 7883118547162052/912673)
(43962454864/72361, 15330199360434795/19465109)
(266065563970/91809, 138352727920706477/27818127)

Commentary

Generic-rank-4 Burkhardt fibration associated with quartic coordinates (5, 11, 74, -40, 130), specialized at t=-8054/3131. PARI/GP 2.17.4 returned rank bounds [12,12].

last edited by jesper-petersen at 2026-10-30 22:12:12 UTC · history