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

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