Models of some genus one curves with applications to descent

Let p denote a prime, and K a field of characteristic prime to p and containing the pth roots of unity. For p equal to 3 and 5, the author finds a scheme T_p and a family of genus one curves over T_p such that any genus one curve defined over the field K of index p whose Jacobian elliptic curve E has is isomorphic to a curve lying over a K-point of T_p. The author then relates the explicit presentation of such families to the program of descent on elliptic curves.