j-invariant determines the moduli of algebraic curves of genus 1
Algebra and Number Theory are the two oldest fundamental branches of mathematics that are at the very center of mathematics even today. Algebra includes the study of structures of solution-sets of algebraic equations, structure of permutations, combinations and transformations. If the defining rules of the structure are commutative then the subject is called Commutative Algebra. Algebraic Geometry investigates the solution-sets of polynomial equations in one or more unknowns; this investigation is aided by the study of abstract structures such as varieties, schemes, stacks etc. Solution-sets of power-series equations also arise naturally in Algebraic Geometry. Number Theory is mainly the study of integers especially, prime numbers. Integer or rational solutions of a system of polynomial equations in one or more unknowns is a part of number theory that goes by the name of Arithmetic Geometry. In Arithmetic Geometry too, certain power-series, called Modular Forms, arise very naturally and play an important role in solving unfathomable mysteries. Algebraic structures such as Fundamental-group and its representations as a group of transformations constitute an important tool in studying symmetries of interesting geometric objects and shapes. Modern era research topics like mathematics of encryption and theory of error correcting codes yield commonly used important real life applications of Algebra and Number Theory: computers, cell-phones, dvds, secured communication on internet all employ Algebra and Number Theory in substantial ways; here beauty of Algebra and Number Theory finds its utility.