  We have performed a comprehensive parameter study of the collapse of
rotating, strongly magnetized stellar cores in axisymmetry to determine their
gravitational wave signature based on the Einstein quadrupole formula. We use a
Newtonian explicit magnetohydrodynamic Eulerian code based on the relaxing-TVD
method for the solution of the ideal MHD equations, and apply the
constraint-transport method to guarantee a divergence--free evolution of the
magnetic field. We neglect effects due to neutrino transport and employ a
simplified equation of state. The pre--collapse initial models are polytropes
in rotational equilibrium with a prescribed degree of differential rotation and
rotational energy (~ 1 % of the gravitational energy). The initial magnetic
fields are purely poloidal the field strength ranging from 10^10 G to 10^13 G.
The evolution of the core, whose collapse is initiated by reducing the gas
pressure by a prescribed amount, is followed until a few ten milliseconds past
core bounce.
  The initial magnetic fields are amplified mainly by the differential rotation
of the core giving rise to a strong toroidal field component. The poloidal
field component grows by compression during collapse, but does not change
significantly after core bounce if (abbreviated)
