TDCOSMO XXVIII. The Hubble constant from the quadruply lensed quasar J1537$-$3010 with precise time delays
We present the first measurement of the Hubble constant ($H_0$) from the quadruply-lensed quasar J1537$-$3010, which has optically-measured time delays at $\sim2 \%$ precision. We combine these delays with multi-band imaging data from the Hubble Space Telescope (HST) and model the system with two independent software and teams. We adopt a mass profile that is maximally degenerate with $H_0$ to fully incorporate the mass-sheet degeneracy in the error budget, with nuisance parameters constrained by spatially resolved stellar kinematics from the Multi Unit Spectroscopic Explorer (MUSE) and a line-of-sight (LoS) analysis using the Euclid Flagship simulation. The entire analysis is performed blindly to $H_0$, distances and mass density slope of the main deflector. After unblinding, we measure $H_0 = 75.5^{+9.3}_{-5.8}\ {\rm km\,s^{-1}\,Mpc^{-1}}$, corresponding to a $10\%$ precision measurement from a single system. This precision is driven by conservative lens modeling assumptions including differences between lens modeling methods and the limited stellar kinematics constraints, from which we infer a total mass-sheet parameter $\lambda\equiv(1-\kappa_{\rm ext})\lambda_{\rm int} = 0.89^{+0.11}_{-0.06}$ ($\lambda_{\rm int}=0.89^{+0.09}_{-0.07}$) that is consistent with current results for elliptical galaxies ($\lambda \approx 1$). Our lensing constraints, stellar kinematic measurements and LoS characterization will be included in subsequent population-level measurements of $H_0$. Moreover, our lens models combined with future near-infrared spectroscopy and imaging from the James Webb Space Telescope will further reduce the uncertainties on $H_0$ from J1537$-$3010 alone, bringing it closer to the few percents precision of the time delays.