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We construct a subfamily of elliptic curves E(r, s) : (y− s)(y+ s) = x(x− r)(x+ r) with r = 2(m− 2m+2m+2m+1), s = 2(m− 2m− 1)(m+1), and show that its rank is at least five over Q(m). This improves the previous results on the rank of the curves E(r, s) over Q(m).

A family of elliptic curves of rank ≥ 5 over Q(m)