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curve #3782

y² = x³ − x² − 113610497864400x + 464397424962142449600

a-invariants: [0, -1, 0, -113610497864400, 464397424962142449600]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 15309447130320
discriminant (Δ): 682983580100349285542160077663970002534400
Faltings height: 6.8817
naive height: 108.7050
primes of bad reduction: 2, 3, 5, 7, 19, 37, 41, 61, 71, 73
regulator: 37916.0491187461911938987879136195729147294469670
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 19:49:13 UTC
last updated: 2026-10-01 19:49:13 UTC

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

(199800, 21016803360)
(9187210, 14002810950)
(856061165/121, 5110497494450/1331)
(15207522040/3249, 1098290433140000/185193)
(38166800821240/4255969, 113384641286347648800/8780064047)

Commentary

(u,v) = (4331,779) 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 19:49:13 UTC · history

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