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We present a strong lensing analysis of the galaxy cluster SDSS J1029+2623 at$z=0.588$, one of the few currently known lens clusters with multiple images ofa background ($z=2.1992$) quasar with a measured time delay. We use archivalHubble Space Telescope multi-band imaging and new Multi Unit SpectroscopicExplorer follow-up spectroscopy to build an accurate lens mass model, a crucialstep towards future cosmological applications. The spectroscopic data enablethe secure identification of 57 cluster members and of two nearby perturbersalong the line-of-sight. We estimate the inner kinematics of a sub-set of 20cluster galaxies to calibrate the scaling relations parametrizing the sub-halomass component. We also reliably determine the redshift of 4 multiply imagedsources, provide a tentative measurement for one system, and report thediscovery of a new four-image system. The final catalog comprises 26 multipleimages from 7 background sources, spanning a wide redshift range, from 1.02 to5.06. We present two parametric lens models, with slightly different clustermass parametrizations. The observed positions of the multiple images areaccurately reproduced within approximately $0''.2$, the three image positionsof the quasar within only $\sim0''.1$. We estimate a cluster projected totalmass of $M(<300~ {\rm kpc}) \sim 2.1 \times 10^{14}~ M_{\odot}$, with astatistical uncertainty of a few percent. Both models, that include a smallgalaxy close to one of the quasar images, predict magnitude differences andtime delays between the quasar images that are consistent with theobservations.

VLT/MUSE observations of SDSS J1029+2623: towards a high-precision strong lensing model