  EK Eri is one of the most slowly rotating active giants known, and has been
proposed to be the descendant of a strongly magnetic Ap star. We have performed
a spectropolarimetric study of EK Eri over 4 photometric periods with the aim
of inferring the topology of its magnetic field. We used the NARVAL
spectropolarimeter at the Bernard Lyot telescope at the Pic du Midi
Observatory, along with the least-squares deconvolution method, to extract high
signal-to-noise ratio Stokes V profiles from a timeseries of 28 polarisation
spectra. We have derived the surface-averaged longitudinal magnetic field Bl.
We fit the Stokes V profiles with a model of the large-scale magnetic field and
obtained Zeeman Doppler images of the surface magnetic strength and geometry.
Bl variations of up to about 80 G are observed without any reversal of its
sign, and which are in phase with photometric ephemeris. The activity
indicators are shown to vary smoothly on a timescale compatible with the
rotational period inferred from photometry (308.8 d.), however large deviations
can occur from one rotation to another. The surface magnetic field variations
of EK Eri appear to be dominated by a strong magnetic spot (of negative
polarity) which is phased with the dark (cool) photometric spot. Our modeling
shows that the large-scale magnetic field of EK Eri is strongly poloidal. For a
rotational axis inclination of i = 60{\deg}, we obtain a model that is almost
purely dipolar. In the dipolar model, the strong magnetic/photometric spot
corresponds to the negative pole of the dipole, which could be the remnant of
that of an Ap star progenitor of EK Eri. Our observations and modeling
conceptually support this hypothesis, suggesting an explanation of the
outstanding magnetic properties of EK Eri as the result of interaction between
deep convection and the remnant of an Ap star magnetic dipole.
