  Given an elliptic curve $E$ defined over $\mathbb{C}$, let $E^{\times}$ be an
open subset of $E$ obtained by removing a point. In this paper, we show that
the $i$-th Betti number of the unordered configuration space
$\mathrm{Conf}^{n}(E^{\times})$ of $n$ points on $E^{\times}$ appears as a
coefficient of an explicit rational function in two variables. We also compute
its Hodge numbers as coefficients of another explicit rational function in four
variables. Our result is interesting because these rational functions resemble
the generating function of the $\mathbb{F}_{q}$-point counts of
$\mathrm{Conf}^{n}(E^{\times})$, which can be obtained from the zeta function
of $E$ over a finite field $\mathbb{F}_{q}$. We show that the mixed Hodge
structure of the $i$-th singular cohomology group
$H^{i}(\mathrm{Conf}^{n}(E^{\times}))$ with complex coefficients is pure of
weight $w(i)$, an explicit integer we provide in this paper. This purity
statement implies our main result about the Betti numbers and the Hodge
numbers. Our proof uses Totaro's spectral sequence computation that describes
the weight filtration of the mixed Hodge structure on
$H^{i}(\mathrm{Conf}^{n}(E^{\times}))$.
