  We split the program of explicit descent of elliptic curves into two parts.
For $n=3$ and $n=5,$ we first display a model for the universal elliptic curve
$E$ with full level $n$ structure and describe the map of rational points of
$E$ to the cohomology group $H^1(G, E[n]).$ Second, we find models in
$\PP^{n-1}$ of principal homogeneous spaces of $E$ corresponding to all
possible elements of $H^1(G, E[n]),$ i.e. for those elements with trivial
period-index obstruction. For this we use the relationship established in
\cite{me2} between the period-index obstruction and the norm symbol, a
generalization of the Hilbert symbol.
