The j-invariant is a mathematical concept used in the field of complex analysis and algebraic geometry. It is a function that classifies elliptic curves over the complex numbers. Specifically, the j-invariant provides a way to distinguish between different elliptic curves, as curves with the same j-invariant are isomorphic, meaning they can be transformed into one another through a change of variables. In addition to its role in classifying elliptic curves, the j-invariant has applications in number theory and cryptography. It is often denoted as j(E) for an elliptic curve E . The j-invariant can also be used in the study of modular forms and has connections to important concepts such as modular arithmetic and Weierstrass equations.