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We present a detailed and pedagogical analysis of recent cosmological data,including CMB, BAO, SnIa and the recent local measurement of $H_0$. We thusobtain constraints on the parameters of standard dark energy parameterizations,including $\Lambda CDM$, and $H(z)$ deformation models such as $wCDM$ (constantequation of state $w$ of dark energy), and the CPL model (corresponding to theevolving dark energy equation-of-state parameter $w(z) = w_0 + w_a\frac{z}{1+z}$). The fitted parameters include the dark matter density$\Omega_{0m}$, the SnIa absolute magnitude $M$, the Hubble constant $H_0$ andthe dark energy parameters (e.g., $w$ for $wCDM$). All models considered leadto a best-fit value of $M$ that is inconsistent with the locally determinedvalue obtained by Cepheid calibrators ($M$ tension). We then use the best-fitdark energy parameters to reconstruct the quintessence Lagrangian that would beable to reproduce these best-fit parameterizations. Due to the derived latephantom behavior of the best-fit dark energy equation-of-state parameter$w(z)$, the reconstructed quintessence models have a negative kinetic term andare therefore plagued with instabilities.

The Hubble Tension, The $M$ Crisis of Late Time $H(z)$ Deformation Models and the Reconstruction of Quintessence Lagrangians