I am trying to get an intuitive idea for the genus of an algebraic curve -- this seems like kind of a common question, but I am having trouble building intuition from past answers. I'm looking specifically at this curve: $$ (x^2 + y^2 - 1)(x^2 + y^2 - 4) = 0 $$ The curve itself is just two concentric circles of radius $1$ and $2$, as shown below. My vague undertstanding of topological genus as "number of holes" tells me that the genus should be $2$, since there are two disconnected components with one "hole" each. But the genus-degree formula seems to be telling me something different; the polynomial has degree $4$, and as far as I can tell has no singular points, so it seems to me like its genus should be $(4-1)(4-2)/2 = 3$. Where have I gone wrong? (I would expect that I have the wrong intuition for algebraic genus, or that there are singular points in projective coordinates that I am not considering.)
Graph generated by Wolfram|Alpha.