curve #6370

y² + xy + y = x³ − 84710269917185528x + 25483744736042717226480006

a-invariants: [1, 0, 1, -84710269917185528, 25483744736042717226480006]
c-invariants: [4066092956024905321, -22017955458040047117716083381]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/6ℤ
conductor (N): 725155197682921070
discriminant (Δ): -241646523498000986859943224700285024021673754751000000
Faltings height: 8.9291
naive height: 130.5233
primes of bad reduction: 2, 5, 13, 31, 53, 79, 131, 421, 443, 1759
regulator: 82643.084387927289631061174537090836649723915080777
submitted by: The Wren
submitted at: 2026-11-01 16:24:32 UTC
last updated: 2026-11-01 16:24:32 UTC

Witness: 6 independent points log in to add more points →

(-229241343, 5732116274259)
(113843345, 4161131487907)
(-926752600357/9604, 5387040370112308291/941192)
(96764254640/361, 32197037014616623/6859)
(2771591849035/324, 4611570941071114559/5832)
(3880919764320/361, 7642670553078755798/6859)

Commentary

Z/6-torsion Kubert–Tate family curve from a rational-parameter search of y^2+(1-c)xy-by=x^3-bx^2, with b=c(c+1), following the construction of David Renshaw on curve#1192 and the example of Lewis Combes on curve#5970. The completed rank-6 batch identified the specialization with raw model [64089,476741095,20873632103480,0,0], reducing to the displayed global minimal model. PARI/GP returned rank bounds [6,6]. The six displayed points were checked exactly on the minimal model. The minimal discriminant was fully factored, and the listed bad-reduction primes are complete.

last edited by The Wren at 2026-11-01 16:24:32 UTC · history