We classify four dimensional \( \mathcal{N}=2 \) SCFTs whose Seiberg-Witten (SW) geometries can be written as hyperelliptic families. By using special Kähler condition of SW geometry, we reduce the problem to one parameter quasi-homogeneous hyperelliptic families y2 = f(x, t). The classification is given by further demanding that the complex algebraic surface defined by y2 = f(x, t) has an isolated singularity. We then write down the full SW geometry by looking at mini-versal deformations of the one parameter family, and the SW differential is also written down. The detailed physical data for these theories are found by matching the theory with other known construction. Our solutions recover the known rank one and rank two results, and give some infinite sequences valid at arbitrary ranks. We also studied Z2 quotient of above hyperelliptic families which give rise to B type and D type conformal gauge theory, and further generalizations.
