Geometry of The Modular Curve
The classical modular curve, which we will call X0(n), is of degree greater than or equal to 2n when n>1, with equality if and only if n is a prime. The polynomial Φn has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with the curve can be difficult. As a polynomial in x with coefficients in Z, it has degree ψ(n), where ψ is the Dedekind psi function. Since Φn(x, y) = Φn(y, x), X0(n) is symmetrical around the line y=x, and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when n>2, there are two singularites at infinity, where x=0, y=∞ and x=∞, y=0, which have only one branch and hence have a knot invariant which is a true knot, and not just a link.
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