Let Q be an algebraic closure of Q, and for any prime
number p, denote by Q(u_p)
the cyclotomic subfield of Q generated by
the pth roots of unity.

Theorem.Let p be a prime. If there exists an elliptic curve E over
Q(u_{p}) such that the points of order p
of E(Q) are all
Q(u_p)-rational, then p=2,3,5,13 or p>1000.