On the rank of elliptic curves over \(\mathbb Q(\sqrt{-3})\) with torsion groups \(\mathbb Z/3\mathbb Z\times\mathbb Z/3\mathbb Z\) and \(\mathbb Z/3\mathbb Z\times \mathbb Z/6\mathbb Z\) (Q719909)

The author constructs elliptic curves over the field \(\mathbb Q(\sqrt{-3})\) with torsion group \(\mathbb Z/3\mathbb Z\times\mathbb Z/3\mathbb Z\) and ranks equal to 7 and an elliptic curve over the same field with torsion group \(\mathbb Z/3\mathbb Z\times \mathbb Z/6\mathbb Z\) and rank equal to 6. This is an improvement of results of \textit{F. P. Rabarison} [Acta Arith. 144, No. 1, 17--52 (2010; Zbl 1228.11085)] who constructed elliptic curves with these torsion groups and ranks \(\geq 2\), \(\geq 3\), respectively.