  In "Proving that a genus 2 curve has complex multiplication", van Wamelen
lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication
(CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois
over $\mathbf{Q}$. The generic case of non-Galois quartic CM-fields did not
feature in this list, as the field of definition in that case always contains a
real quadratic field, known as the real quadratic subfield of the reflex field.
  We extend van Wamelen's list to include curves of genus two defined over this
real quadratic field. Our list therefore contains the smallest "generic"
examples of CM curves of genus two.
  We explain our methods for obtaining this list, including a new
height-reduction algorithm for arbitrary hyperelliptic curves over totally real
number fields. Unlike Van Wamelen, we also give a proof of our list, which is
made possible by our implementation of denominator bounds of Lauter and Viray
for Igusa class polynomials.
