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curve #3781

y² + xy + y = x³ + x² − 9381673137660x + 2119317475465193565

a-invariants: [1, 1, 1, -9381673137660, 2119317475465193565]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 13843123266030
discriminant (Δ): 50906726379913066956705911155137538560000
Faltings height: 6.5012
naive height: 101.2229
primes of bad reduction: 2, 3, 5, 17, 23, 41, 43, 67, 97, 103
regulator: 7134.19322187735100811436981706351975168246267961
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 19:48:50 UTC
last updated: 2026-10-01 19:48:50 UTC

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

(-2431947, 3249543473)
(-459107, 2516126873)
(-1000732611/361, 17926842209115/6859)
(301137925/49, 4556850839687/343)
(1077771373/81, 34443892363697/729)

Commentary

(u,v) = (4945,3296) 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 19:48:50 UTC · history

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