William Stein's Publications: The field generated by the points of small prime order on an elliptic curve
The field generated by the points of small prime order on
an elliptic curve
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.
Let p be a prime. If there exists an elliptic curve E over
Q(up) 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.
The paper: pdf,
There is also a page with some
software and tables related to the paper.