We discuss a new method to bound 5-torsion in class groups using elliptic curves. The most natural ``trivial'' bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a ``dimension 0 selmer group'', and by embedding it into an appropriate Elliptic curve we can bound its size by the Tate-Shafarevich group which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds ``dimension 0 selmer groups'' by embedding into global objects, and using class number formulas.