Class groups of imaginary quadratic points on $X_1(16)$

The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ is an imaginary quadratic field and $E$ is an elliptic curve over $K$ with a torsion point of order 16, then the class number of $K$ is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to $X_1(16)$.