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curve #4051

y² = x³ − 16335512067x + 651672509502274

a-invariants: [0, 0, 0, -16335512067, 651672509502274]
c-invariants: [784104579216, -563045048209964736]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 10859531085360
discriminant (Δ): 95522826469059113081126976000000
Faltings height: 4.8528
naive height: 82.1634
primes of bad reduction: 2, 3, 5, 47, 61, 83, 241, 263
regulator: 962.89742178132225561524146375635320203987929171869
submitted by: Michael Rubinstein
submitted at: 2026-10-07 22:24:44 UTC
last updated: 2026-10-07 22:24:44 UTC

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

(-13417, 29469150)
(379505, 221608008)
(492057/4, 179543525/8)
(-139657, 14462370)
(26033, -15622200)
(-131767, 22723200)

Commentary

Torsion Z/2+Z/2. From Michael Rubinstein's github repository: https://github.com/mrubinst/elliptic-curves-by-rank

last edited by Michael Rubinstein at 2026-10-07 22:24:44 UTC · history

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