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curve #3776

y² = x³ − x² − 485212422378104x + 3684222217232867898624

a-invariants: [0, -1, 0, -485212422378104, 3684222217232867898624]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 163259706516816
discriminant (Δ): 1447233266225010531397101310736385087528059904
Faltings height: 7.3977
naive height: 113.0604
primes of bad reduction: 2, 3, 7, 13, 17, 19, 29, 53, 79, 953
regulator: 920.668900374163353098215477523788076407193714714
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 19:47:06 UTC
last updated: 2026-10-01 19:47:06 UTC

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

(-18091191, 80877716556)
(4230654, 41318015466)
(6919340, 25654464108)
(21622504, 57462521984)
(7267789/4, 423970951495/8)

Commentary

(u,v) = (6409,2597) 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 19:47:06 UTC · history

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