  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.
