Abstract. Let E/Q be an elliptic curve and let Q(3 ∞ ) be the compositum of all cubic extensions of Q. In this article we show that the torsion subgroup of E(Q(3 ∞ )) is finite and determine 20 possibilities for its structure, along with a complete description of the Q-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many Q-isomorphism classes of elliptic curves, and a complete list of j-invariants for each of the 4 that do not.

Let E/Q be an elliptic curve and let Q(D ∞ 4 ) be the compositum of all extensions of Q whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of D4. In this article we first show that Q(D ∞ 4 ) is infact the compostium of all D4 extensions of Q and then we prove that the torsion subgroup of E(Q(D ∞ 4 )) is finite and determine the 24 possibilities 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.

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Given an elliptic curve E/Q with torsion subgroup G = E(Q)tors we study what groups (up to isomorphism) can occur as the torsion subgroup of E base-extended to K, a degree 6 extension of Q. We also determine which groups H = E(K)tors can occur infinitely often and which ones occur for only finitely many curves. This article is a first step towards a complete classification of torsion growth over sextic fields.

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