On The Elliptic Curve Y <sup> 2 </ Sup> = X (x + ¦Òp) (x + ¦Òq) In The Class Number <sub> 1 </ Sub> Imaginary Quadratic Fields Of Selmer Groups And Mordell-weil, Group Structure,

Consider a family of elliptic curves E = E_σ: y² = x(x +σp)(x +σq),whereσ=±1,p and q are prime numbers with q-2 = p. In this paper,we will give out explicitly the following several parameters of E over nine imaginary quadratic number fields K = Q((?)) with class number one where D = -3,-4,-7, -8, -11, -19,-43,-67,-163i)the Selmer groups S^(?)(E/K),S^(?)(E'/K). e.g.S((?))(E₊/K) = (Z/2Z)² when p≡31(mod 56) and K = Q((?)),etc.ii)the sum of rankE(K) and the 2-,(?)-, (?)- parts of Shafarevich-Tate groups Sha(E/K). e.g.rank(E₊(K))+dim₂(sha(E₊/K[(?)])+dim₂(sha(E'₊/K[(?)])= 3,whenp≡31(mod 56) and K =Q((?)).iii)Mordell-Weil goup E(K).e.g.E₊(K)≌Z/2Z×Z/2Z, Sha(E₊/K)[2] = 0, when K = Q((?)) and p≡5(mod 8).