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We describe the simplest non-trivial modular category $\mathfrak{E}$ with twosimple objects. Then we extract from this category the invariant fornon-oriented links in 3-sphere and two invariants for 3-manifolds: thecomplex-valued Turaev - Reshetikhin type invariant $tr_{\varepsilon}$ and thereal-valued Turaev - Viro type invariant $tv_{\varepsilon}$. These twoinvariants 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.

Invariants for links and 3-manifolds from the modular category with two simple objects