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We study elliptic curves in Weierstrass form (with integer coefficients) that have noninteger torsion points over Q. After putting the curves in certain normal forms, we find conditions on their coefficients characterizing when the torsion group is isomorphic to ℤ/2ℤ x ℤ/2Nℤ, for each N = 1, 2, 3, 4.

Construction of elliptic curves with noninteger torsion points and noncyclic torsion groups