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curve #3784

y² + xy + y = x³ + x² − 64566360259204x + 41243023074393192821

a-invariants: [1, 1, 1, -64566360259204, 41243023074393192821]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 381938061851346
discriminant (Δ): 16491745683400607498201812815489584137660944
Faltings height: 6.9831
naive height: 107.0097
primes of bad reduction: 2, 3, 7, 11, 13, 19, 29, 31, 47, 113, 701
regulator: 1417.05343716620610513823448816775437751661026018
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 20:39:20 UTC
last updated: 2026-10-01 20:39:20 UTC

Witness: 5 independent points log in to add more points →

(-5749723, 14915951325)
(449327, 3510093147)
(592967, 1778971797)
(55169707, 405433865397)
(-10334371/25, 1029759532857/125)

Commentary

(u,v) = (8008,5311) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 5.

last edited by Daniel Błażewicz at 2026-10-01 20:39:20 UTC · history

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