  In projectivized strata of meromorphic $1$-forms on elliptic curves with only
one zero, the locus of residueless differentials is a complex curve endowed
with a canonical complex projective structure. Drawing on the multi-scale
compactification of strata, we provide formulas to compute the genus of these
curves and the degree of their natural forgetful map to $\mathcal{M}_{1,1}$.
Additionally, we distinguish two non-hyperelliptic components in the
residueless locus of the exceptional stratum $\mathcal{H}_{1}(12,-3,-3,-3,-3)$
and hence complete the classification of the connected components of these
loci.
