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Reference.org
Classical modular curve
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<p>In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the classical modular curve is an irreducible <a href="/facts/Algebraic_curve/LnWsPmDP">plane algebraic curve</a> given by an equation </p> Φ<i>n</i>(<i>x</i>, <i>y</i>) = 0, <p>such that (<i>x</i>, <i>y</i>) = (<i>j</i>(<i>nτ</i>), <i>j</i>(<i>τ</i>)) is a point on the curve. Here <i>j</i>(<i>τ</i>) denotes the <a href="/facts/J-invariant/H1a38AJF">j-invariant</a>. </p><p>The curve is sometimes called <i>X</i>0(<i>n</i>), though often that notation is used for the abstract <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> for which there exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as Φ<i>n</i>(<i>x</i>, <i>x</i>). </p><p>The classical modular curves are part of the larger theory of <a href="/facts/Modular_curve/UE3kT5SW">modular curves</a>. In particular it has another expression as a compactified quotient of the complex <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a> H. </p>