Context. During the performance verification phase of the Spectrum-Roentgen-Gamma (SRG) eROSITA telescope, the eROSITA Final Equatorial-Depth Survey (eFEDS) was carried out. It covers a 140 deg2 field located at 126◦ < R.A. < 146◦
and -3◦ < Dec. < +6◦ with a nominal unvignetted exposure over the field of 2.2 ks. Five hundred and fourty-two candidate clusters and groups were
detected in this field, down to a flux limit FX ∼ 10−14 erg s−1 cm−2 in the 0.5-2 keV band.
Aims. In order to understand radio-mode feedback in galaxy clusters, we study the radio emission of brightest cluster galaxies (BCGs) of eFEDS clusters and groups, and we relate it to the X-ray properties of the host cluster.
Methods. Using LOFAR, we identified 227 radio galaxies hosted in the BCGs of the 542 galaxy clusters and groups detected in eFEDS. We treated non-detections as radio upper limits. We analysed the properties of radio galaxies, such as redshift and luminosity distribution, offset from the cluster centre, largest linear size, and radio power. We studied their relation to the intracluster medium of
the host cluster.
Results. We find that BCGs with radio-loud active galactic nucleus (AGN) are more likely to lie close to the cluster centre than radioquiet BCGs. There is a clear relation between the cluster X-ray luminosity and the 144 MHz radio power of the BCG. Statistical tests indicate that this correlation is not produced by biases or selection effects in the radio band. We see no apparent link between largest linear size of the radio galaxy and the central density in the host cluster. Converting the radio luminosity into kinetic luminosity, we find that radiative losses of the intracluster medium are in an overall balance with the heating provided by the central AGN. Finally, we tentatively classify our objects into disturbed and relaxed based on different morphological parameters, and we show that the link between the AGN and the ICM apparently holds for both subsamples, regardless of the dynamical state of the cluster.