  M87 is a giant elliptical galaxy located in the centre of the Virgo cluster,
which harbours a supermassive black hole of mass 6.4x10^9 M_sun, whose activity
is responsible for the extended (80 kpc) radio lobes that surround the galaxy.
The energy generated by matter falling onto the central black hole is ejected
and transferred to the intra-cluster medium via a relativistic jet and
morphologically complex systems of buoyant bubbles, which rise towards the
edges of the extended halo. Here we present the first observations made with
the new Low-Frequency Array (LOFAR) of M87 at frequencies down to 20 MHz.
Images of M87 were produced at low radio frequencies never explored before at
these high spatial resolution and dynamic range. To disentangle different
synchrotron models and place constraints on source magnetic field, age and
energetics, we also performed a detailed spectral analysis of M87 extended
radio-halo using these observations together with archival data. We do not find
any sign of new extended emissions; on the contrary the source appears well
confined by the high pressure of the intra-cluster medium. A continuous
injection of relativistic electrons is the model that best fits our data, and
provides a scenario in which the lobes are still supplied by fresh relativistic
particles from the active galactic nuclei. We suggest that the discrepancy
between the low-frequency radio-spectral slope in the core and in the halo
implies a strong adiabatic expansion of the plasma as soon as it leaves the
core area. The extended halo has an equipartition magnetic field strength of
~10 uG, which increases to ~13 uG in the zones where the particle flows are
more active. The continuous injection model for synchrotron ageing provides an
age for the halo of ~40 Myr, which in turn provides a jet kinetic power of
6-10x10^44 erg/s.
