Torsion subgroups of elliptic curves in elementary abelian 2-extensions of Q

Let E be an elliptic curve over Q and let F≔Q({m;m∈Z}). Laska and Lorenz showed that there exist at most 31 possibilities for the type of the torsion subgroup E(F)tors of E over F. In this paper, we showed that there exist exactly 20 possibilities for E(F)tors.