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Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi-$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) andtheir dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x(x \pm p) (x \pm q) $ over imaginary quadratic number fields of class numberone are determined explicitly in many cases.

On the Selmer groups and Mordell-Weil groups of elliptic curves $ y^{2}  = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class  number one