We have performed detailed temporal and time-integrated spectral analysis of 286 bursts from SGR J1550-5418 detected with the Fermi Gamma-ray Burst Monitor (GBM) in January 2009, resulting in the largest uniform sample of temporal and spectral properties of SGR J1550-5418 bursts. We have used the combination of broadband and high time-resolution data provided with GBM to perform statistical studies for the source properties. We determine the durations, emission times, duty cycles and rise times for all bursts, and find that they are typical of SGR bursts. We explore various models in our spectral analysis, and conclude that the spectra of SGR J1550-5418 bursts in the 8-200 keV band are equally well described by optically thin thermal bremsstrahlung (OTTB), a power law with an exponential cutoff (Comptonized model), and two black-body functions (BB+BB). In the spectral fits with the Comptonized model we find a mean power-law index of -0.92, close to the OTTB index of -1. We show that there is an anti-correlation between the Comptonized Epeak and the burst fluence and average flux. For the BB+BB fits we find that the fluences and emission areas of the two blackbody functions are correlated. The low-temperature BB has an emission area comparable to the neutron star surface area, independent of the temperature, while the high-temperature blackbody has a much smaller area and shows an anti-correlation between emission area and temperature. We compare the properties of these bursts with bursts observed from other SGR sources during extreme activations, and discuss the implications of our results in the context of magnetar burst models.
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