In the paper On the structure of conic bundles. Math. USSR, Izv., 120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}(X,\mathbb{Z})=0$.
The construction starts by taking an irreducible curve $C \subset \mathbb{P}^2$ whose normalisation has genus one, with $\deg(C) > 12$ and ODP singularities. Then taking a minimal map of surfaces $S \rightarrow \mathbb{P}^2$ which resolves the singularities of $C$. Then taking a conic bundle $X$ over $S$ with discriminant curve equal to the pull back of $C$.
My question is, what is the minimal value of $b_{2}(X)$ which can be given by this construction? For this minimal value is there an explicit description of a corresponding curve $C$?
