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The paper formulates a precise relationship between the Tate-Shafarevichgroup of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of theclassgroup 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.

Relating Tate-Shafarevich group of an elliptic curve with class group