Algebraic torsion is a means of understanding the topological complexity of certain homomorphic curves counted in some Floer theories of contact manifolds.  This talk focuses on algebraic torsion and the contact invariant in embedded contact homology, which has largely been left unexplored over the past decade, but can be used to understand symplectic fillability and overtwistedness of the contact 3-manifold.  We restrict to concave linear plumbings, whose boundaries are contact lens spaces, and explain the curve counting methods we have developed and results that we have obtained, which parallels with the contact toric description of these lens spaces. This talk is based on joint work in progress with Aleksandra Marinkovic, Ana Rechtman, Laura Starkston, Shira Tanny, and Luya Wang.  We are exploring the use of our methods to better understand nonfillable tight contact 3-manifolds obtained from more general plumbings.