An elliptic curve $E$ over $\mathbb{ Q}$ is said to be good if $N_{ E}^{ 6}<\max\!\left\{ \left\vert c_{ 4}^{ 3}\right\vert ,c_{ 6}^{ 2}\right\} $ where $N_{ E}$ is the conductor of $E$ and $c_{ 4}$ and $c_{ 6}$ are the invariants associated to a global minimal model of $E$. In this article, we generalize Masser's Theorem on the existence of infinitely many good elliptic curves with full $2$-torsion. Specifically, we prove via constructive methods that for each of the fifteen torsion subgroups $T$ allowed by Mazur's Torsion Theorem, there are infinitely many good elliptic curves $E$ with $E\!\left(\mathbb{ Q}\right) _{ \text{ tors}}\cong T$.
Categories: math.NT