In this paper,we consider a quadratic twist family of the elliptic curve 14a2.Using the classical 2-descent method,we prove that all curves in this family have 2-Selmer group(Z/2Z)2.Assume the finiteness of the Shafarevich-Tate group,then these curves have rank 1,Selmer Z2-corank 1,and trivial 2-primary part of the Shafarevich-Tate group.According to the Gross-Zagier formula and the Tunnell-Saito theorem,we raise the question of how to prove the non-triviality of certain explicit Heegner points defined on Shimura curves associated to the quaternion algebra precisely ramified at 2 and 7.
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