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curve #3740

y² + xy + y = x³ + x² − 16078420003833145x + 784429849769968003133870

a-invariants: [1, 1, 1, -16078420003833145, 784429849769968003133870]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 357067097216445
discriminant (Δ): 194776482947709644937899434183038474938094140625
Faltings height: 8.0349
naive height: 123.5624
primes of bad reduction: 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 1129
regulator: 5970.49571594971053748300544465918846965747434845
submitted by: Daniel Błażewicz
submitted at: 2026-10-01 04:20:54 UTC
last updated: 2026-10-01 04:20:54 UTC

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

(-80916827, 1247293039713)
(304343207/4, 315146523399/8)
(618189218, 15069223861368)
(57051036923/676, 2971446453024213/17576)
(228693897122/4489, 93921800084956344/300763)

Commentary

(u, v)=(29716, 3025) 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 04:24:05 UTC · history

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