A genus two curve related to the class number one problem

We give another solution to the class number one problem via a correspondence of imaginary quadratic fields of class number h(−d)=1h(−d)=1 to integral points on a genus 2 curve K3K3 defined by the equation 8x8−32x6y+40x4y2+64x5−16x2y3−128x3y+y4+48xy2+96x2−32y−24=08x8−32x6y+40x4y2+64x5−16x2y3−128x3y+y4+48xy2+96x2−32y−24=0. In fact one can find all rational points on K3K3. Besides the integral points related to the integral points on Heegner's elliptic curve K2:y2=2x(x3−1)K2:y2=2x(x3−1), the curve K3K3 has four more (rational) points: (−3,6)(−3,6), (1,2)(1,2), (−917,6289), (−15579,424866241). Our approach is a further step of exploiting Weber's conjecture (Birch's theorem after [1]) on special values of Schläfli–Weber modular functions. It involves also a genus 9 curve K1K1 and its sub-covering K1→KsK1→Ks studied independently by Heegner and Stark.