  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 two between the ROSAT HRI and ASCA
observations, which were about one year apart. The X-ray spectrum of the quasar
is fit 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 fit 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 ICM emission.
The radial surface distribution of the emission was not fit by either the beta
model which 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 NNW and SSE, 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.
