  An accurate knowledge of the scattering length is fundamental in ultracold
quantum gas experiments and essential for the characterisation of the system as
well as for a meaningful comparison to theoretical models. Here, we perform a
careful characterisation of the s-wave scattering length $a_s$ for the four
highest-abundance isotopes of erbium, in the magnetic field range from 0G to
5G. We report on cross-dimensional thermalization measurements and apply the
Enskog equations of change to numerically simulate the thermalization process
and to analytically extract an expression for the so-called number of
collisions per re-thermalization (NCPR) to obtain $a_s$ from our experimental
data. We benchmark the applied cross-dimensional thermalization technique with
the experimentally more demanding lattice modulation spectroscopy and find good
agreement for our parameter regime. Our experiments are compatible with a
dependence of the NCPR with $a_s$, as theoretically expected in the case of
strongly dipolar gases. Surprisingly, we experimentally observe a dependency of
the NCPR on the density, which might arise due to deviations from an ideal
harmonic trapping configuration. Finally, we apply a model for the dependency
of the background scattering length with the isotope mass, allowing to estimate
the number of bound states of erbium.
