Generators and number fields for torsion points of a special elliptic curve

Abstract

Let E be an elliptic curve with Weierstrass form y2=x3−px, where p is a prime number and let E[m] be its m-torsion subgroup. Let p1=(x1,y1) and p2=(x2,y2) be a basis for E[m], then we prove that Q(E[m])=Q(x1,x2,ξm,y1) in general. We also find all the generators and degrees of the extensions Q(E[m])∕Q for m=3 and m=4.