Lin et al. 2020 bring us a new phylogeny
of the Eosauropterygia (defined as: Pachypleurosaurus (Fig. 1) and descendants, or Sauropterygia sans Placodontia).

Figure 1. Pachypleurosaurus had more than two sacrals all converging on a tiny ilium.
From the Lin et all 2020 abstract:
“During the last two decades, abundant Triassic sauropterygians have been reported from Europe and southwestern China, which greatly improve our understanding of the diversity and stratigraphic, as well as paleogeographic, distribution of Triassic sauropterygians. The phylogeny of Sauropterygia was also repeatedly analyzed with the description of each new species. Except for Placodontia, however, various analyses of sauropterygian interrelationships have yielded incongruent results, especially with regards to the monophyly of Pachypleurosauridae and Eusauropterygia (= Eosauropterygia sans Pachypleurosauridae) which were alternatively supported or rejected by different analyses.”
“The incongruent results of these analyses were probably caused by the implementation of different character codings, based primarily on the same data matrix taken from a global parsimony analysis by Rieppel about 20 years ago, which was based almost exclusively on European material. Since we have a large number of new specimens from the rest of the world, it is high time to reexamine the phylogeny of the entire group, or at least of the eosauropterygians, in order to assess whether and how the newly described taxa affect overall tree topology.”

Figure 2. Aquatic younginiform subset of the LRT demonstrating relationships within the Enaliosauria (=Sauropterygia + Ichthyosauria)
Continuing from the Lin et all 2020 abstract:
“Given that the ingroup relationships of Placodontia is well established, we herein focus on reanalyzing the interrelationships of Eosauropterygia, which is the sister clade of the Placodontia within the Sauropterygia. We present a comprehensive phylogenetic hypothesis for Eosauropterygia based on a cladistic analysis of 137 characters coded for four outgroup and 49 ingroup taxa, including nearly all currently recognized Triassic eosauropterygian genera. This is the most inclusive phylogenetic analysis of Eosauropterygia to date.”
Does it include the taxa listed above (Fig. 1). If not, add them.
“The new phylogenetic hypothesis of Eosauropterygia suggests that Pachypleurosauridae is the sister taxon of Eusauropterygia, and their monophyly as traditionally upheld is reestablished.”
Which taxon is the last common ancestor? In the large reptile tree (LRT, 1751+ taxa, subset Fig. 2) indicates that Eusauropterygia is indeed monophyletic and arises from a series of pachypleurosaurs. This is distinct from the Lin et al. results which recovers a monophyletic Pachypleurosauridae.
“Furthermore, the monophyly of the genus Nothosaurus as traditionally conceived is not supported, whereas the monophyly of Lariosaurus is obtained if the lariosaurian affinity of N. juvenilis, N. youngi, and N. winkelhorsti is accepted. In this study, the monophyletic Pistosauroidea excludes Corosaurus and Cymatosaurus. The latter two genera are found to form a clade that represents the basal-most members of Eusauropterygia. The new phylogenetic hypothesis is mostly in good accordance with the stratigraphic distribution of the genera.”
I cannot comment further on a cladogram I have not seen. Overall the LRT recovers a traditional cladogram, with the exception of a short list of non-tradtional outgroups, like the Mesosauria, Thalattosauria and Ichthyosauria along with a long list of basal aquatic diapsids. The clade Enaliosauria is traditionally considered obsolete, but the LRT recovers it as a monophyletic clade. Turtles are not closely related when the valid, tested turtle sisters and ancestors are included in the taxon list. The initial radiation of sauropterygians evidently occurred prior to the Early Triassic given the diversity present in the Early Triassic and the diversity of mesosaurs in the Late Permian.
References
Lin W, Jiang D, Rieppel O, Motany R, Tintori A, Sun Z and Zhou M 2020. Phylogeny of the Eosauropterygia (Diapsida: Sauropterygia) incorporating recent discoveries from South China. SVP Abstracts 2020.