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ABSTRACT Covering $$\sim 5600\, \deg ^2$$ to rms sensitivities of ∼70−100 $$\mu$$Jy beam−1, the LOFAR Two-metre Sky Survey Data Release 2 (LoTSS-DR2) provides the largest low-frequency (∼150 MHz) radio catalogue to date, making it an excellent tool for large-area radio cosmology studies. In this work, we use LoTSS-DR2 sources to investigate the angular two-point correlation function of galaxies within the survey. We discuss systematics in the data and an improved methodology for generating random catalogues, compared to that used for LoTSS-DR1, before presenting the angular clustering for ∼900 000 sources ≥1.5 mJy and a peak signal-to-noise ≥ 7.5 across ∼80 per cent of the observed area. Using the clustering, we infer the bias assuming two evolutionary models. When fitting angular scales of $$0.5 \le \theta \lt 5{^\circ }$$, using a linear bias model, we find LoTSS-DR2 sources are biased tracers of the underlying matter, with a bias of $$b_{\rm C}= 2.14^{+0.22}_{-0.20}$$ (assuming constant bias) and $$b_{\rm E}(z=0)= 1.79^{+0.15}_{-0.14}$$ (for an evolving model, inversely proportional to the growth factor), corresponding to $$b_{\rm E}= 2.81^{+0.24}_{-0.22}$$ at the median redshift of our sample, assuming the LoTSS Deep Fields redshift distribution is representative of our data. This reduces to $$b_{\rm C}= 2.02^{+0.17}_{-0.16}$$ and $$b_{\rm E}(z=0)= 1.67^{+0.12}_{-0.12}$$ when allowing preferential redshift distributions from the Deep Fields to model our data. Whilst the clustering amplitude is slightly lower than LoTSS-DR1 (≥2 mJy), our study benefits from larger samples and improved redshift estimates. 

We report the discovery of a complete Einstein ring around the elliptical galaxy NGC 6505, atz = 0.042. This is the first strong gravitational lens discovered inEuclidand the first in an NGC object from any survey. The combination of the low redshift of the lens galaxy, the brightness of the source galaxy (I_E = 18.1 lensed,I_E = 21.3 unlensed), and the completeness of the ring make this an exceptionally rare strong lens, unidentified until its observation byEuclid. We present deep imaging data of the lens from theEuclidVisible Camera (VIS) and Near-Infrared Spectrometer and Photometer (NISP) instruments, as well as resolved spectroscopy from theKeckCosmic Web Imager (KCWI). TheEuclidimaging in particular presents one of the highest signal-to-noise ratio optical/near-infrared observations of a strong gravitational lens to date. From the KCWI data we measure a source redshift ofz = 0.406. Using data from the Dark Energy Spectroscopic Instrument (DESI) we measure a velocity dispersion for the lens galaxy ofσ_⋆ = 303 ± 15 km s⁻¹. We model the lens galaxy light in detail, revealing angular structure that varies inside the Einstein ring. After subtracting this light model from the VIS observation, we model the strongly lensed images, finding an Einstein radius of 2.″5, corresponding to 2.1 kpc at the redshift of the lens. This is small compared to the effective radius of the galaxy,R_eff ∼ 12.″3. Combining the strong lensing measurements with analysis of the spectroscopic data we estimate a dark matter fraction inside the Einstein radius off_DM = (11.1_−3.5^+5.4)% and a stellar initial mass-function (IMF) mismatch parameter ofα_IMF = 1.26_−0.08^+0.05, indicating a heavier-than-Chabrier IMF in the centre of the galaxy.