  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$).
