curve #2165

y² = x³ + x² − 1063298052122346851354580x + 418289670457782230165570156300729184

a-invariants: [0, 1, 0, -1063298052122346851354580, 418289670457782230165570156300729184]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 294986299695893757693379720933488412787544
discriminant (Δ): 1353313441060786486966131853463742562426551872215180871005207459445129472
Faltings height: 12.6585
naive height: 177.5839
primes of bad reduction: 2, 3, 7, 37, 41, 43, 61, 127, 163, 179, 199, 239, 277, 283, 607, 1097, 17047
regulator: 360238448883.1915061719904595963833477668928484999174471829121187
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:17:36 UTC
last updated: 2026-09-26 14:17:36 UTC

Witness: 13 independent points log in to add more points →

(901172968614, 438096152073226902)
(778671108060, 249921656585530272)
(222507029454, 438992669778857262)
(315585665016, 337873622441105016)
(1375220545071, 1247750304897931866)
(1962945007614, 2427884205308227902)
(211548730056, 450352894580771616)
(663336961710, 69595961180265522)
(917647487535, 463989920651450022)
(762524981574, 225530891982612342)
(-12145301853521700/18769, 2350201771063996954741584/2571353)
(4171211844707256/3481, 190984840514455092185736/205379)
(20416896231772974/24025, 1334432866710684334315482/3723875)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=-9/13; one of the smallest fibres of the family (|a|,b<=40 over 8 values of u), rank certified with PARI ellrank: 13 independent points.

last edited by Steps Unbounded at 2026-09-26 14:17:36 UTC · history