  Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi
-$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) and
their 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 number
one are determined explicitly in many cases.
