The field generated by the points of small prime order on an elliptic curve

Loic Merel and William A. Stein

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(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.

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