Explicit classification for torsion cyclic subgroups of rational points with even orders of elliptic curves

For elliptic curves E over the rationals Q, the classification according to their torsion subgroups E-tors(Q) of rational points has been studied. Whe n E-tors(Q) are cyclic groups with even orders, the classification is given with explicit critria, and the generators of the torsion groups are also e xplicitly presented in each case. These results, together with the recent r esults of Ono for the non-cyclic torsion groups, have completely solved the problem of the explicit classification with E being a rational point of or der 2.