Long-duration gamma-ray bursts (GRBs) have been found to be associated with broad-lined type-Ic supernovae (SNe), but only a handful of cases have been studied in detail. Prompted by the discovery of the exceptionally bright, nearby GRB130427A (redshift z=0.3399), we aim at characterising the properties of its associated SN2013cq. This is the first opportunity to test directly the progenitors of high-luminosity GRBs. We monitored the field of the Swift long duration GRB130427A using the 3.6-m TNG and the 8.2-m VLT during the time interval between 3.6 and 51.6 days after the burst. Photometric and spectroscopic observations revealed the presence of the type Ic SN2013cq. Spectroscopic analysis suggests that SN2013cq resembles two previous GRB-SNe, SN1998bw and SN2010bh associated with GRB980425 and XRF100316D, respectively. The bolometric light curve of SN2013cq, which is significantly affected by the host galaxy contribution, is systematically more luminous than that of SN2010bh ($sim$ 2 mag at peak), but is consistent with SN1998bw. The comparison with the light curve model of another GRB-connected SN2003dh, indicates that SN2013cq is consistent with the model when brightened by 20%. This suggests a synthesised radioactive $^{56}$Ni mass of $sim 0.4 M_odot$. GRB130427A/SN2013cq is the first case of low-z GRB-SN connection where the GRB energetics are extreme ($E_{rm gamma, iso} sim 10^{54}$ erg). We show that the maximum luminosities attained by SNe associated with GRBs span a very narrow range, but those associated with XRFs are significantly less luminous. On the other hand the isotropic energies of the accompanying GRBs span 6 orders of magnitude (10$^{48}$ erg $< E_{rm gamma, iso} <$ 10$^{54}$ erg), although this range is reduced when corrected for jet collimation. The GRB total radiated energy is in fact a small fraction of the SN energy budget.
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