I have some affine planar quartic curves over $\mathbb{R}$ (of the general form $a x^4 + b x^3y + c x^2 y^2 + d xy^3 + e y^4 + fx^3 + g x^2y + h xy^2 + i y^3 + j x^2 + k xy + l y^2 + m x + n y + p= 0$ with $(a,b,c,d,e,f,g,h,i,j,k,l, m,n,p)\in \mathbb{R}^{15}$) that I know for sure describes either:
- The empty set
- 1 point
- 2 points
- 3 points
- 4 points
This is the case because it is the sum of the square of two different conics, so it describes the intersection which cannot be a full overlap.
I would like to count those points (using a closed form if possible).
I thought about finding some invariant describing the number of connected component of the curve or something like that (maybe the number of bitangent too, knowing that not every quartic get 28 bitangents), but I could not find a reference listing the known invariants of the planar quartics and describing them.
Does anyone have a reference of that sort?