Abstract
Let $E$ be an elliptic curve defined over $\mathbb{Q} $. We study the relationship between the torsion subgroup $E(mathbb{Q} )_{tors}$ and the torsion subgroup $E(K)_{tors}$, where $K$ is a cubic number field. In particular, we study the number of cubic number fields $K$ such that $E(\mathbb{Q} )_{tors}\neq E(K)_{tors}$.
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