  We study families of elliptic curves of degree n+1 in $P^n$ containing a
fixed set of m points. In the case m = n+3 we show that this family is
birationally isomorphic to a smooth complete intersection of n-2 diagonal
quadrics in $P^{n+2}$. We also describe an action of the 2-elementary abelian
group $2^{n+2}$ on this family via Cremona transformations in $P^n$.
