In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation
- Φn(x, y)=0,
where for the j-invariant j(τ),
- x=j(n τ), y=j(τ)
is a point on the curve. The curve is sometimes called X0(n), though often that is used for the abstract algebraic curve for which there exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x).
It is important to note that the classical modular curves are part of the larger theory of modular curves. In particular it has another expression as a compactified quotient of the complex upper half-plane H.
Read more about Classical Modular Curve: Geometry of The Modular Curve, Parametrization of The Modular Curve, Mappings, Galois Theory of The Modular Curve
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