Algebraic numbers and elliptic curves

The values of the modular j-invariant at CM points are called singular moduli. They determine the isomorphism classes of the corresponding CM elliptic curves,
and have been known to be algebraic integers since the time of Kronecker and Weber. This connection between algebra and geometry has produced lots of beautiful
mathematics. In this talk, we will take a look at some of them concerning factorizations of these algebraic numbers, and what they tell us about isogenies of CM
elliptic curves.