The truth about torsion in the CM case, II

Let $T_{\mathrm{C}\mathrm{M}}(d)$ be the largest size of the torsion subgroup of an elliptic curve with complex multiplication (CM) defined over a degree $d$ number field. Work of Breuer and Clark–Pollack showed $\limsup_{d \to \infty} \frac{T_{{\rm CM}}(d)}{d \log \log d} \in (0,\infty)$. Here we show that the above limit supremum is precisely $\frac{e^{\gamma }\pi }{\sqrt{3}}$. We also study – in part, out of necessity – the upper order of the size of the torsion subgroup of various restricted classes of CM elliptic curves over number fields.