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We construct an elliptic curve over Q(i) with torsion group Z/4Z * Z/4Z and rank equal to 7 and a family of elliptic curves with the same torsion group and rank >= 2.

On the rank of elliptic curves over $\mathbf{Q}(i)$ with torsion group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/4\mathbf{Z}$