  (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.
