curve #2162

y² = x³ − x² − 7865618818149775897740x + 267341454488481158401019215375716

a-invariants: [0, -1, 0, -7865618818149775897740, 267341454488481158401019215375716]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 16305275978428710457836900037023245758488
discriminant (Δ): 268638460991272988217079322143830763883844104458994380173970205952
Faltings height: 11.4019
naive height: 162.8640
primes of bad reduction: 2, 3, 7, 23, 29, 83, 109, 139, 149, 151, 179, 193, 337, 389, 1493, 2179, 3023, 115469
regulator: 891978093910.1004093953858943049354119214570359617516264499725200
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:17:25 UTC
last updated: 2026-09-26 14:17:25 UTC

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

(55602654876, 1377100091401074)
(-71448228612, 21554432050780818)
(54045575208, 320296107012462)
(44628960438, 2279590425653868)
(-76780089440, 20460492645477654)
(2909436479644/49, 1065168631473771138/343)
(99203703264, 21525421222361898)
(-89826516016, 15782589431083802)
(48228019884, 416900491537698)
(-222630956/9, 441626269440877442/27)
(1710647676163602/27889, 18427438836769083168276/4657463)
(53764935347624/961, 46164683548407507118/29791)
(173734689425310738/1852321, 47480031389486563966337148/2521008881)

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=29/13, t=-1/7; 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 14:17:26 UTC · history