(shortened) The SMC is ideally suited to investigating the recent star formation history from X-ray source population studies. It harbours a large number of Be/X-ray binaries, and the supernova remnants can be easily resolved with imaging X-ray instruments. We search for new supernova remnants in the SMC and in particular for composite remnants with a central X-ray source. We study the morphology of newly found candidate supernova remnants using radio, optical and X-ray images and investigate their X-ray spectra. Here we report on the discovery of the new supernova remnant around the recently discovered Be/X-ray binary pulsar SXP 1062 in radio and X-ray images. The Be/X-ray binary system is found near the centre of the supernova remnant, which is located at the outer edge of the eastern wing of the SMC. The remnant is oxygen-rich, indicating that it developed from a type Ib event. From XMM-Newton observations we find that the neutron star with a spin period of 1062 s shows a very high average spin-down rate of 0.26 s per day over the observing period of 18 days. From the currently accepted models, our estimated age of around 10000-25000 years for the supernova remnant is not long enough to spin down the neutron star from a few 10 ms to its current value. Assuming an upper limit of 25000 years for the age of the neutron star and the extreme case that the neutron star was spun down by the accretion torque that we have measured during the XMM-Newton observations since its birth, a lower limit of 0.5 s for the birth spin period is inferred. For more realistic, smaller long-term average accretion torques our results suggest that the neutron star was born with a correspondingly longer spin period. This implies that neutron stars in Be/X-ray binaries with long spin periods can be much younger than currently anticipated.
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