  Optical time series photometry of the short period magnetic white dwarf +
probable brown dwarf binary SDSS 121209.31+013627.7 reveals pulse-like
variability in all bands from i' to u', peaking at u'. These modulations are
most likely due to a self-eclipsing accretion hot spot on the white dwarf,
rotating into view every 88.43 minutes. This period is commensurate with the
radial velocity period determined by Schmidt et al. 2005 of ~90 minutes, and
consistent with the rotation period of the accretor being equal to the binary
orbital period. We combine our observations with those recently published by
Koen and Maxted 2006 to provide an accurate ephemeris. We also detect the
system in X-rays with Swift, and estimate the accretion rate at ~1x10^-13Msun
per year. We suggest that SDSS1212 is most likely a magnetic cataclysmic
variable in an extended state of very low accretion, similar to the
well-studied Polar EF Eri. Alternatively, the putative brown dwarf is not
filling its Roche Lobe and the system is a detached binary in which the white
dwarf is efficiently accreting from the wind of the secondary. Six such
post-common envelope, ``pre-Polar'' systems - termed ``low accretion rate
Polars (LARPs)'' by Schwope et al. 2002 - have previously been identified
through optical cyclotron emission lines. Cyclotron emission from SDSS1212 has
recently been detected in the near-IR Debes et al. 2006 but, if detached, it
would be the first ``LARP'' with a probably sub-stellar secondary. It is
unclear whether an L-dwarf wind is strong enough to provide the measured
accretion rate. We suggest further observations to distinguish between the
Roche Lobe over-flow and wind accretion scenarios.
