On the rank of elliptic curves over $\mathbf{Q}(\sqrt{-3})$ with torsion groups $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/3\mathbf{Z}$ and $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/6\mathbf{Z}$

Abstract

We construct elliptic curves over the field $\\mathbf{Q}(\\sqrt{-3})$ with torsion group $\\mathbf{Z}/3\\mathbf{Z} \\times \\mathbf{Z}/3\\mathbf{Z}$ and ranks equal to 7 and an elliptic curve over the same field with torsion group $\\mathbf{Z}/3\\mathbf{Z} \\times \\mathbf{Z}/6\\mathbf{Z}$ and rank equal to 6.