  OB stars are important constituents for the ecology of the Universe, and
there are only a few studies on their pulsational properties detailed enough to
provide important feedback on current evolutionary models. Our goal is to
analyse and interpret the behaviour present in the CoRoT light curve of the
B0.5 IV star HD 51756 observed during the second long run of the space mission,
and to determine the fundamental stellar parameters from ground-based
spectroscopy gathered with the CORALIE and HARPS instruments after checking for
signs of variability and binarity, thus making a step further in mapping the
top of the Beta Cep instability strip. We compare the newly obtained
high-resolution spectra with synthetic spectra of late O-type and early B-type
stars computed on a grid of stellar parameters. We match the results with
evolutionary tracks to estimate stellar parameters. We use various time series
analysis tools to explore the nature of the variations present in the light
curve. Additional calculations are carried out based on distance and historical
position measurements of the components to impose constraints on the binary
orbit. We find that HD 51756 is a wide binary with both a slow (v sin i \approx
28 km s^-1) and a fast (v sin i \approx 170 km s^-1) early-B rotator whose
atmospheric parameters are similar (T_eff \approx 30000 K and log g \approx
3.75). We are unable to detect pulsation in any of the components, and we
interpret the harmonic structure in the frequency spectrum as sign of
rotational modulation, which is compatible with the observed and deduced
stellar parameters of both components. The non-detection of pulsation modes
provides a feedback on the theoretical treatment, given that non-adiabatic
computations applied to appropriate stellar models predict the excitation of
both pressure and gravity modes for the fundamental parameters of this star.
