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curve #5508

y² + xy + y = x³ + x² − 10997951183380x − 5984976891803584348

a-invariants: [1, 1, 1, -10997951183380, -5984976891803584348]
c-invariants: [527901656802241, 5171016075255870859711]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 50758437763005
discriminant (Δ): 69662216746432430372914828694232486350625
Faltings height: 6.5311
naive height: 101.6998
primes of bad reduction: 3, 5, 7, 29, 47, 109, 1063, 3061
regulator: 5221.98569206334470796246492248765592902221092905
submitted by: Daniel Błażewicz
submitted at: 2026-10-22 20:04:26 UTC
last updated: 2026-10-22 20:04:26 UTC

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

(-599149, 624280974)
(-562348, 148149636)
(3608432, 1144618491)
(-16832000/9, 76593485596/27)
(935745437/16, 28576055441649/64)

Commentary

(u,v) = (5123,4060) 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 2026-10-22 20:04:26 UTC · history

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