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FormClassNumber ​

How many classes of indefinite binary quadratic forms a discriminant has. A class is not one reduced form but a CYCLE of them, and that cycle is a periodic continued fraction — which is to say, a closed geodesic.

WikipediaClass number (number theory)MathWorldClassNumberWikidataQ91783763nLabclass number
FormClassNumber(D)the number of classes
FormCycle(form)the class the form lies in, listed
ReducedForms(D)every reduced form of that discriminant
FormAutomorph(form)the hyperbolic matrix generating the form's stabiliser
PellSolution(D)the fundamental with

Domain: The modular group

Details
  • Indefinite means and is not a perfect square; a square discriminant gives a form that factors, with no cycle
  • acts by and preserves , so classes live inside one discriminant
  • Gauss's reduction condition is , and `FormRho` steps round the cycle
  • A cycle's length is always even, because flips the sign of the leading coefficient
  • The automorph has trace , hence is hyperbolic — so a form class IS one of the closed geodesics ModularClasses counts
  • A discriminant has no forms at all, and the class number is 0

Examples

See also: ModularClasses, ContinuedFraction, RademacherSymbol