Abstract
We study the classification of elliptic curves E over the rationals ℚ according to the torsion sugroups E tors(ℚ). More precisely, we classify those elliptic curves with E tors(ℚ) being cyclic with even orders. We also give explicit formulas for generators of E tors(ℚ). These results, together with the recent results of K. Ono for the non-cyclic E tors(ℚ), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.
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