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curve #3752

y² + xy = x³ − 3986123229x + 56575621948824

a-invariants: [1, 0, 0, -3986123229, 56575621948824]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 7827180207
discriminant (Δ): 2670756009454461650004352469169
Faltings height: 4.5376
naive height: 77.9319
primes of bad reduction: 3, 7, 11, 19, 37, 157, 307
regulator: 1381.011461728219792345481221782675434072182947
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 08:21:13 UTC
last updated: 2026-10-01 08:21:13 UTC

Witness: 4 independent points log in to add more points →

(-51711, 11180499)
(6600, 5524332)
(10343619, 33260837049)
(8507532/841, 101281325706/24389)

Commentary

(u,v) = (703,396) for the C2 x C4 family y^2 = x(x+u^2)(x+v^2). PARI/GP gives exact rank 4.

last edited by Daniel Błażewicz at 2026-10-01 08:21:13 UTC · history

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