The properties of the brightest Lyman alpha emitters at z~5.7

(Abridged) We use deep VLT optical and near-IR spectroscopy and deep Spitzer/IRAC imaging to examine the properties of two of the most luminous Lyman alpha emitters at z=5.7. The continuum red-ward of the Lyman alpha line is clearly detected in both objects, thus facilitating a relatively accurate measurement (10--20% uncertainties) of the observed rest-frame equivalent widths, which are around 160 Angstroms for both objects. Through detailed modelling of the profile of the Lyman alpha line with a 3-D Monte-Carlo radiative transfer code, we estimate the intrinsic rest-frame equivalent width of Lyman alpha and find values that are around 300 Angstroms, which is at the upper end of the range allowed for very young, moderately metal-poor star-forming galaxies. However, the uncertainties are large and values as high as 700 Angstroms are permitted by the data. Both Lyman alpha emitters are detected at 3.6 micron in deep images taken with the Spitzer Space Telescope. We use these measurements, the measurement of the continuum red-ward of Lyman alpha and other photometry to constrain the spectral energy distributions of these very luminous Lyman alpha emitters and to compare them to three similar Lyman alpha emitters from the literature. The contribution from nebular emission is included in our models: excluding it results in significantly higher masses. Evidence for dust is found in all objects, and emission from nebular lines often make a dominant contribution to the rest frame 3.6 micron flux. Rich in emission lines, these objects are prime targets for the next generation of extremely large telescopes, JWST and ALMA.