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Proposition 1 Let E: y 2 = F(x) = x 3 + ax + b be an elliptic curve defined over Fp having three rational 2-torsion points. If p ≡ 3 mod 4, there are no rational 4-torsion points on E. Remember Swan’s theorem [3]. Let () a p denote the Legendre’s symbol. Theorem 2 Let f(X) be a squarefree polynomial of degree d and n its number of irreducible factors modulo p. The

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