Good elliptic curves with a specified torsion subgroup

Abstract

An elliptic curve $E$ over $\mathbb{Q}$ is said to be good if $N_E^6<\max \left\{\left|c_4^3\right|,c_6^2\right\}$ 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 $E{\left(\mathbb{Q}\right)}_{\text{tors}}\cong T$.

19 pages; incorporates referee's suggestions; final version to appear in Journal of Number Theory

References in corpus (2)

• A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves
• Minimal models of rational elliptic curves with non-trivial torsion

Cited by in corpus (1)

• Lower bounds for the modified Szpiro ratio