We report the detection of eight bright X-ray bursts from the 6.5-s magnetar 1E 1048.1-5937, during a 2013 July observation campaign with the Nuclear Spectroscopic Telescope Array (NuSTAR). We study the morphological and spectral properties of these bursts and their evolution with time. The bursts resulted in count rate increases by orders of magnitude, sometimes limited by the detector dead time, and showed blackbody spectra with kT=6-8 keV in the T90 duration of 1-4 s, similar to earlier bursts detected from the source. We find that the spectra during the tail of the bursts can be modeled with an absorbed blackbody with temperature decreasing with flux. The bursts flux decays followed a power-law of index 0.8-0.9. In the burst tail spectra, we detect a ~13 keV emission feature, similar to those reported in previous bursts from this source as well as from other magnetars observed with the Rossi X-ray Timing Explorer (RXTE). We explore possible origins of the spectral feature such as proton cyclotron emission, which implies a magnetic field strength of B~2X10^15 G in the emission region. However, the consistency of the energy of the feature in different objects requires further explanation.
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