  We bound the running time of an algorithm that computes the genus-two class
polynomials of a primitive quartic CM-field K. This is in fact the first
running time bound and even the first proof of correctness of any algorithm
that computes these polynomials.
  Essential to bounding the running time is our bound on the height of the
polynomials, which is a combination of denominator bounds of Goren and Lauter
and our own absolute value bounds. The absolute value bounds are obtained by
combining Dupont's estimates of theta constants with an analysis of the shape
of CM period lattices.
  The algorithm is basically the complex analytic method of Spallek and van
Wamelen, and we show that it finishes in time Otilde(Delta^(7/2)), where Delta
is the discriminant of K. We give a complete running time analysis of all parts
of the algorithm, and a proof of correctness including a rounding error
analysis. We also provide various improvements along the way.
