Torsion subgroups of rational elliptic curves over the compositum of all extensions of generalized \(D_4\)-type

Abstract

Let \(E/\mathbb{Q}\) be an elliptic curve and let \(\mathbb{Q}(D_4^\infty)\) be the compositum of all extensions of \(\mathbb{Q}\) whose Galois closure has Galois group isomorphic to a subdirect product of a finite number of transitive subgroups of \(D_4\). In this article we prove that the torsion subgroup of \(E(\mathbb{Q}(D_4^\infty))\) is finite and determine the 24 possibility for its structure. We also give a complete classification of the elliptic curves that have each possible torsion structure in terms of their \(j\)-invariants.