On elliptic curves $ y^{2} = x(x + \varepsilon p )(x + \varepsilon q) $ and their twists

Abstract: In this paper, we study the arithmetic of elliptic curves $ y^{2} = x(x + \varepsilon p )(x + \varepsilon q) $ and their twists. The following arithmetic quantities are explicitly determined: the norm index $ \delta (E, \Q, K), $ the root numbers, the set of anomalous prime numbers, a few prime numbers at which the image of Galois representation are surjective. Some results about the relation between the ranks of the Mordell-Weil groups, Selmer groups and Shafarevich-Tate groups, and the structure about the $ l^{\infty }-$Selmer groups and the Mordell-Weil groups over $ \Z_{l}-$extension via Iwasawa theory are obtained. The parity conjecture for some of these curves are confirmed.