$$\mathcal F(\lambda, y) = {y}^{4}- \left( 2 {\lambda}^{8}+{\lambda}^{4}+2 \right) {y}^{3}+ \left( 2 {\lambda}^{16}+4 {\lambda}^{14}+3 {\lambda}^{12} +4 {\lambda}^{2}+2 \right) {y}^{2} \\-4 {\lambda}^{2}\left( {\lambda}^{20}+ {\lambda}^{16}+4 \right) y+{\lambda}^{4}(4 {\lambda}^{24}-3 {\lambda}^{20}+4 )=0$$ I am trying to construct the Riemann theta function solution of a soliton equation. The curve comes from the characteristic polynomial of the Lax pair of the soliton equation.
Noting that the curve is singular at the original point $(0, 0)$, I want to know if it is possible to make the curve a compact Riemann surface. If so, what is its genus, the local coordinates at the origin?