curve #1262

y² + xy + y = x³ − 6444653414828197028043x + 199134938077524109763523334648358

a-invariants: [1, 0, 1, -6444653414828197028043, 199134938077524109763523334648358]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/6ℤ
conductor (N): 140409740846986028110 ☆ record for ℤ/6ℤ torsion, rank ≥ 7
discriminant (Δ): 273659691134861160738979381654867205404992276906910250000 ☆ record for ℤ/6ℤ torsion, rank ≥ 7
Faltings height: 10.8655
naive height: 162.2662
primes of bad reduction: 2, 5, 7, 11, 17, 19, 23, 179, 271, 2677, 189019
regulator: 609249.4926380451040735741905617645480661395365311854
submitted by: hdeping
submitted at: 2026-09-24 16:46:49 UTC
last updated: 2026-09-24 16:46:49 UTC

(46348730619, -56843481560)
(185401645511/4, 4399887178205/8)
(63511842895411/1369, 830067123525905820/50653)
(44541537512587/961, 2231624780594680/29791)
(26697024292537/576, 59827525889353/13824)
(619926485344, 484197125540039490)
(22476151923724249/4096, -3369277733330160695634365/262144)

Found by a Mestre-Nagao sieve over the Tate normal form family y^2+txy+(t+2)y=x^3 with torsion Z/6Z, at t=745873/3401.

last edited by hdeping at 2026-09-27 08:24:40 UTC · history