The canonical module of a quasihomogeneous normal affine $SL_2$-variety

For the varieties mentioned in the title a description is given for the canonical divisor, the Picard group, and the divisor class group. In particular, it follows from this that the singular 3-dimensional quasihomogeneous $SL_2$-varieties are not Gorenstein. The canonical module is described. All descriptions are given in terms of discrete parameters: the height and the degree of a quasihomogeneous affine $SL_2$-variety.