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curve #13

y² = x³ − 82x

a-invariants: [0, 0, 0, -82, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
conductor (N): 430336
naive height: 24.8338
Faltings height: 0.1377
discriminant (Δ): 35287552
primes of bad reduction: 2, 41
regulator: 10.207892029767887379642364292744681701782086
submitted by: David Renshaw
submitted at: 2026-05-27 22:11:57 UTC
last updated: 2026-07-01 23:04:07 UTC

Witness: 3 independent points

(-8, 12)
(-1, 9)
(49/4, 231/8)

Commentary

Found by Billing (1938). A curve of rank exactly 3, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:45 UTC · history

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