Birational transformations of threefold $\mathbf{Q}$-conic bundles

A $\mathbf{Q}$-conic bundle is a contraction $f:X\to Z$ of a three-dimensional algebraic variety $X$ to a surface~$Z$ such that the variety~$X$ has only terminal $\mathbf{Q}$-factorial singularities, the anticanonical divisor $-K_X$ is~$f$-ample, and $\text{uprho}(X\mathrm{/}Z)=1$. We provide an algorithm to transform a $\mathbf{Q}$-conic bundle to its standard form.