A Genus Two Curve Related to the Class Number One Problem

We give another solution to the class number one problem by showing that imaginary quadratic fields $\text{Q}(\sqrt{-d})$ with class number $h(-d)=1$ correspond to integral points on a genus two curve ${\text{mscrK}}_3$. In fact one can find all rational points on ${\text{mscrK}}_3$. The curve ${\text{mscrK}}_3$ arises naturally via certain coverings of curves:\ ${\text{mscrK}}_3\text{rg}{\text{mscrK}}_6$,
${\text{mscrK}}_1\text{rg}{\text{mscrK}}_2$\ with ${\text{mscrK}}_2\text{\hspace{0.17em}⁣}:y^2=2x(x^3-1)$ denoting the Heegner curve, also in connection with the so-called Heegner-Stark covering ${\text{mscrK}}_1\text{rg}{\text{mscrK}}_s$.