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curve #3778

y² = x³ + x² − 98207449485384x + 270762881650073370756

a-invariants: [0, 1, 0, -98207449485384, 270762881650073370756]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 275622260785872
discriminant (Δ): 28948604427520257275570820615827630107198464
Faltings height: 7.0470
naive height: 108.2679
primes of bad reduction: 2, 3, 7, 11, 13, 29, 31, 47, 127, 1069
regulator: 1414.72956352711143236040878284757987380007020727
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 19:47:52 UTC
last updated: 2026-10-01 19:47:52 UTC

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

(2918030, 3006216444)
(8792096, 9324753438)
(151078856, 1853046551202)
(-1841259694/361, 173382010778592/6859)
(-75989990/9, 602552117728/27)

Commentary

(u,v) = (4371,2233) 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 19:47:52 UTC · history

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