X-Ray Properties of B2 1028+313: A Quasar at the Center of the Abell Cluster A1030

X-ray observations with the ROSAT HRI and with ASCA are presented for the nearby radio quasar B2 1028+313, which is located in the cD galaxy at the center of the Abell cluster A1030. We also analyze archival ROSAT PSPC observations. We find that the X-ray emission is dominated by the quasar. The flux varied by a factor of about 2 between the ROSAT HRI and ASCA observations, which were about 1 yr apart. The X-ray spectrum of the quasar is fitted by a single power law, except at low energies where there is a soft excess. Although the shape of the soft excess is not strongly constrained, it can be fitted by a blackbody with a temperature of about 30 eV. There was evidence for extended X-ray emission, which contributed about 25% of the total flux. However, this emission does not appear to be normal X-ray emission from intracluster gas or a central cooling flow. The extended X-ray emission appears to be quite soft; if its spectrum is modeled as thermal emission, the temperature is ~0.2 keV, rather than the 5-10 keV expected for intracluster medium (ICM) emission. The radial surface distribution of the emission was not fitted by either the beta model that usually describes ICM emission or by a cooling-flow model. The ASCA and ROSAT spectra showed no convincing evidence for a thermal component with a cluster-like temperature, either in the overall spectral shape or in emission lines. In addition, the ROSAT PSPC image showed that the extended X-ray emission was highly elongated to the north-northwest and the south-southeast, in the same direction as the extended radio emission from the quasar. We suggest that the extended emission is inverse Compton emission from the extended radio lobes.