On sup-norm bounds part II: $GL(2)$ Eisenstein series

In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for $GL(2)$ over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maa{\ss} forms. Our results generalise those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field $\mathbb{Q}$ had been considered.