Fields generated by torsion points of elliptic curves

Let K be a field of characteristic char(K)≠2,3 and let E be an elliptic curve defined over K. Let m be a positive integer, prime with char(K) if char(K)≠0; we denote by E[m] the m -torsion subgroup of E and by K_m:=K(E[m]) the field obtained by adding to K the coordinates of the points of E[m]. Let P_i:=(x_i,y_i) (i=1,2) be a Z-basis for E[m]; then K_m=K(x_1,y_1,x_2,y_2). We look for small sets of generators for K_m inside {x_1,y_1,x_2,y_2,ζ_m} trying to emphasize the role of ζ_m (a primitive m -th root of unity). In particular, we prove that K_m=K(x_1,ζ_m,y_2), for any odd m>3. When m=p is prime and K is a number field we prove that the generating set {x_1,ζ_p,y_2} is often minimal, while when the classical Galois representation Gal(K_p/K)→GL2(Z/pZ) is not surjective we are sometimes able to further reduce the set of generators. We also describe explicit generators, degree and Galois groups of the extensions K_m/K for m=3 and m=4.