The preferentially magnified active nucleus in IRAS F10214+4724-III. VLBI observations of the radio core

We report 1.7 GHz very long baseline interferometry (VLBI) observations of IRAS F10214+4724, a lensed z = 2.3 obscured quasar with prodigious star formation. We detect what we argue to be the obscured active nucleus with an effective angular resolution of < 50 pc at z = 2.3. The S-1.7 = 210 mu Jy (9 Sigma) detection of this unresolved source is located within the Hubble Space Telescope rest-frame ultraviolet/optical arc, however, greater than or similar to 100 mas northwards of the arc centre of curvature. This leads to a source-plane inversion that places the European VLBI Network detection to within milliarcseconds of the modelled cusp caustic, resulting in a very large magnification (mu similar to 70), over an order of magnitude larger than the CO (1 -> 0) derived magnification of a spatially resolved Jansky Very Large Array (JVLA) map, using the same lens model. We estimate the quasar bolometric luminosity from a number of independent techniques and with our X-ray modelling find evidence that the AGN may be close to Compton thick, with an intrinsic bolometric luminosity of log(10)(<< L-bol,L- QSO >>/L-circle dot) = 11.34 +/- 0.27 dex. We make the first black hole mass estimate of IRAS F10214+4724 and find log(10)(M-BH/M-circle dot) = 8.36 +/- 0.56 which suggests a low black hole accretion rate (lambda = (M)over dot/(M)over dot(Edd) similar to 3 +/-(7)(2) per cent). We find evidence for an M-BH/M-spheroid ratio that is one to two orders of magnitude larger than that of submillimetre galaxies (SMGs) at z similar to 2. At face value, this suggests that IRAS F10214+4724 has undergone a different evolutionary path compared to SMGs at the same epoch. A primary result of this work is the demonstration that emission regions of different sizes and positions can undergo significantly different magnification boosts (> 1 dex) and therefore distort our view of high-redshift, gravitationally lensed galaxies.