curve #5851

y² + y = x³ + 119823152079

a-invariants: [0, 0, 1, 0, 119823152079]
c-invariants: [0, -103527203396472]
rank (lower bound): ≥ 9
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 2067492639485818598152401
discriminant (Δ): -6202477918457455794457203
Faltings height: 3.4361
naive height: 64.5417
primes of bad reduction: 3, 239, 2005408403
regulator: 912282525630.40436073514725685109345167564487545131098788
submitted by: jesper-petersen
submitted at: 2026-10-25 12:06:11 UTC
last updated: 2026-10-25 12:06:11 UTC

Witness: 9 independent points log in to add more points →

(-119823150439/24304900, 14498601465559/119823157000)
(-9593508007/58767556, 155943868489230487/450512084296)
(6127501805137/6423701904, 178860758520955595399/514846860201792)
(3274671722561/473497600, 6916356918002867809/10303307776000)
(359937202915529/42894552100, 7489218010235274864167/8883890685431000)
(19674138218561/1524121600, 89663400211875078559/59501707264000)
(75768797478977/3729789184, 664228069954286083681/227785685045248)
(80286539522561/3876307600, 724223825031926137441/241338911176000)
(10179182501728489/5890911494400, 5054726501086774825905013/14297949106288128000)

Commentary

Burkhardt parametrization (-2,-9,10), generic rank-four sextic; specialization t=-64/109. The 3-isogenous partner has global minimal equation y^2+y=x^3+119823152079. PARI/GP 2.17.4 returned rank bounds [9,9].

last edited by jesper-petersen at 2026-10-25 12:06:11 UTC · history