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 \sim 6–8 keV in the T _ { 90 } 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 \sim 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 \sim 2 \times 10 ^ { 15 } G in the emission region . However , the consistency of the energy of the feature in different objects requires further explanation .