curve #2163

y² = x³ + x² − 16791938577642222790615x + 16079645324828906777789424999290834

a-invariants: [0, 1, 0, -16791938577642222790615, 16079645324828906777789424999290834]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 294986299695893757693379720933488412787544
discriminant (Δ): -111392729501243415563031633154958510364231521565006537693848946846144048
Faltings height: 12.3119
naive height: 171.0489
primes of bad reduction: 2, 3, 7, 37, 41, 43, 61, 127, 163, 179, 199, 239, 277, 283, 607, 1097, 17047
regulator: 320211954562.8368943751026307523407535705714208888155086070329944
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:17:28 UTC
last updated: 2026-09-26 14:17:28 UTC

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

(168020285887, 134170074377566899)
(2896191488165/4, 4954191353809735401/8)
(-8741671424599/400, 1025629381211328462051/8000)
(1148629390931, 1229729855762202711)
(5283524307029/4, 12128580805775893137/8)
(-273750603085, 12717888738012297)
(310777949666, 202180432494494436)
(89201552289286/81, 841716828366538143194/729)
(246423418310, 164029441121687778)
(-1196793476345434/9409, 116012226751373826545814/912673)
(-34211591727223/256, 517086413332365290661/4096)
(343183489382, 225245050840646214)
(96130729442, 123910368220115484)

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 (the 2-isogenous partner of that fibre); 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 14:17:28 UTC · history