We review the main conjecture for an elliptic curve on Q having good supersingular reduction at p and give some consequences of it. Then we define the notion of \lambda -invariant and of \mu - invariant in this situation, generalizing a work of Kurihara and deduce from it the behaviour of the order of the group of Shafarevich-Tate along the cyclotomique Z^p -extension. By examples, we give some arguments which, by allying theorems and numeral calculations, allow to calculate the order of the p -primary part of the group of Shafarevich-Tate in not yet known cases (non trivial Shafarevich-Tate group, curves of rank greater than 1 ).