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In a previous paper, the author examined the possible torsions of an ellipticcurve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$.Although all the possible torsions were found if the elliptic curve hasrational coefficients, we were unable to eliminate some possibilities for thetorsion if the elliptic curve has coefficients that are not rational. In thisnote, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain aclassification of all possible torsions of elliptic curves over $\mathbb Q(i)$and $\mathbb Q(\sqrt{-3})$.

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields