The above observations reflect the fact that $E$ acts transitively by pullback on its own ${\rm Pic}^{b}$ for $b>0$ (for instance, a holomorphic line bundle of degree $1$ on $E$ is uniquely isomorphic to $\mathcal{O}_{E}(x)$ for some point $x\in E$). Moreover, the total spaces of all degree $b>0$ holomorphic line bundles on $E$ are actually biholomorphic as complex manifolds. All of this is false in the classical ALH case $b=0$. In particular, for $b=0$ the above parameters $p,q$ are actual moduli of the metric, corresponding to flat metrics on $\mathbb{T}^{3}$ that do not split isometrically as $S^{1}\times\mathbb{T}^{2}$.