Lower bounds for the modified Szpiro ratio

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $\sigma_{m}(E) =\log\max\left{ \left\vert c_{4}^{3}\right\vert ,c_{6}^{2}\right} /\log N_{E}$where$c_{4}$and$c_{6}$ are the invariants associated to a global minimal model of $E$, and $N_E$ denotes the conductor of $E$. In this article, we show that for each of the fifteen torsion subgroups $T$ allowed by Mazur’s Torsion Theorem, there is a rational number $l_T$ such that if $T\hookrightarrow E(\mathbb{Q}{)}_{\text{tors}}$, then $\sigma_{m}(E)

l_{T}$. We also show that this bound is sharp.