  The paper formulates a precise relationship between the Tate-Shafarevich
group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the
classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) =
GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over
${\mathbb Z}/p$. We establish such a relationship in most cases.
