  We describe the simplest non-trivial modular category $\mathfrak{E}$ with two
simple objects. Then we extract from this category the invariant for
non-oriented links in 3-sphere and two invariants for 3-manifolds: the
complex-valued Turaev - Reshetikhin type invariant $tr_{\varepsilon}$ and the
real-valued Turaev - Viro type invariant $tv_{\varepsilon}$. These two
invariants for 3-manifolds are related by the equality
$|tr_{\varepsilon}|^2\cdot (\varepsilon + 2) = tv_{\varepsilon}$, where
$\varepsilon$ is a root of the equation $\varepsilon^2 = \varepsilon + 1$.
Finally, we show that $tv_{\varepsilon}$ coincides with the well-known
$\varepsilon$ invariant for 3-manifolds.
