LOFAR properties of SILVERRUSH Ly$\alpha$ emitter candidates in the
  ELAIS-N1 field
  Lyman alpha emitters (LAEs) in the Epoch of Reionization (EoR) offer valuable
probes of early galaxy evolution and the process of reionization; however, the
exact evolution of their abundance and the nature of their emission remain open
questions. We combine samples of 229 and 349 LAE candidates at $z=5.7$ and
$z=6.6,$ respectively, from the SILVERRUSH narrowband survey with deep Low
Frequency Array (LOFAR) radio continuum observations in the ELAIS-N1 field to
search for radio galaxies in the EoR and study the low-frequency radio
properties of $z\gtrsim5.7$ LAE emitters. Our LOFAR observations reach an
unprecedented noise level of $\sim20\,\mu$Jy beam$^{-1}$ at 150MHz, and we
detect five candidate LAEs at $>5\sigma$ significance. Based on detailed
spectral energy distribution modelling of independent multi-wavelength
observations, we conclude that these sources are likely [OII] emitters at
$z=1.47$, yielding no reliable $z\gtrsim5.7$ radio galaxy candidates. We
examine the 111 $z=5.7$ and $z=6.6$ LAE candidates from our panchromatic
photometry catalogue that are undetected by LOFAR, finding contamination rates
of 81-92% for the $z=5.7$ and $z=6.6$ subset of the LAE candidate samples. This
subset is biased towards brighter magnitudes and redder near-infrared colours.
The contamination rates of the full sample will therefore likely be lower than
the reported values. Contamination is lowered significantly through constraints
on the near-infrared colours, highlighting the need for infrared observations
to robustly identify bright LAEs in narrowband surveys. Finally, the stacking
of radio continuum observations for the robust LAE samples yields 2$\sigma$
upper limits on radio luminosity of 8.2$\times$10$^{23}$ and
8.7$\times$10$^{23}$ W Hz$^{-1}$ at $z=5.7$ and $6.6$, respectively,
corresponding to limits on their median star-formation rates of $<$53 and $<$56
M$_{\odot}$ yr$^{-1}$.
