  We explicitly construct all supersymmetric flux vacua of a particular
Calabi-Yau compactification of type IIB string theory for a small number of
flux carrying cycles and a given D3-brane tadpole. The analysis is performed in
the large complex structure region by using the polynomial homotopy
continuation method, which allows to find all stationary points of the
polynomial equations that characterize the supersymmetric vacuum solutions. The
number of vacua as a function of the D3 tadpole is in agreement with
statistical studies in the literature. We calculate the available tuning of the
cosmological constant from fluxes and extrapolate to scenarios with a larger
number of flux carrying cycles. We also verify the range of scales for the
moduli and gravitino masses recently found for a single explicit flux choice
giving a K\"ahler uplifted de Sitter vacuum in the same construction.
