Number fields generated by the 3-torsion points of an elliptic curve

摘要

Let ${{mathcal{E}}}$ be an elliptic curve, m a positive number and ${mathcal{E}[m]}$ the m-torsion subgroup of ${mathcal{E}}$ . Let P 1 = (x 1, y 1), P 2 = (x 2, y 2) form a basis of ${mathcal{E}[m]}$ . We prove that ${mathbb Q(mathcal{E}[m]) = mathbb Q(x_1, x_2, zeta_m, y_1)}$ in general. For the case m = 3 we provide a description of all the possible extensions ${mathbb Q(mathcal{E}[3])}$ in terms of generators, degree and Galois groups.