Lower bounds for the modified Szpiro ratio

Let $E/mathbb{Q}$ be an elliptic curve. The modified Szpiro ratio of $E$ is the quantity $sigma_{m}( E) =logmaxleft{ leftvert c_{4}^{3}rightvert ,c_{6}^{2}right} /log N_{E}$ 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 Mazurs Torsion Theorem, there is a rational number $l_{T}$ such that if $Thookrightarrow E(mathbb{Q})_{text{tors}}$, then $sigma_{m}(E) >l_{T}$. We also show that this bound is sharp if the $ABC$ Conjecture holds.