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We present a gravitational wave (GW) analysis of an extensive series of three-dimensional magnetohydrodynamical core-collapse simulations. Our 25 models are based on a 15  progenitor stemming from (i) stellar evolution calculations; (ii) a spherically symmetric effective general relativistic potential, either the Lattimer-Swesty (with three possible compressibilities) or the Shen equation of state for hot, dense matter; and (iii) a neutrino parametrisation scheme that is accurate until about 5 ms postbounce. For three representative models, we also included long-term neutrino physics by means of a leakage scheme, which is based on partial implementation of the isotropic diffusion source approximation (IDSA). We systematically investigated the effects of the equation of state, the initial rotation rate, and both the toroidal and the poloidal magnetic fields on the GW signature. We stress the importance of including of …