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

(t, u)

with

t^2 - Du^2 = 4

Typed signatures

Type	Package	Applies
`(integer) -> integer`	`modular`	always

Domain: The modular group

Details

Indefinite means $D > 0$ and $D$ is not a perfect square; a square discriminant gives a form that factors, with no cycle
$\mathrm{SL}(2,\mathbb{Z})$ acts by $(x,y) \mapsto (px+qy, rx+sy)$ and preserves $D$ , so classes live inside one discriminant
Gauss's reduction condition is $|\sqrt{D} - 2|a|| < b < \sqrt{D}$ , and FormRho steps round the cycle
A cycle's length is always even, because $\rho$ flips the sign of the leading coefficient
The automorph has trace $t > 2$ , hence is hyperbolic — so a form class IS one of the closed geodesics ModularClasses counts
A discriminant $\not\equiv 0, 1 \pmod 4$ has no forms at all, and the class number is 0

Plot

Examples

Basic #

four classes of discriminant 60

In

\mathrm{FormClassNumber}(60)

Out

4

Scope #

the discriminant of a form

In

\mathrm{FormDiscriminant}(\mathrm{QuadraticForm}(1, 1, -1))

Out

5

indefinite when $D=5$

In

\mathrm{IsIndefinite}(\mathrm{QuadraticForm}(1, 1, -1))

Out

\mathrm{True}

definite when $D=-4$

In

\mathrm{IsIndefinite}(\mathrm{QuadraticForm}(1, 0, 1))

Out

\mathrm{False}

already reduced

In

\mathrm{IsReducedForm}(\mathrm{QuadraticForm}(1, 1, -1))

Out

\mathrm{True}

$x^2+xy-y^2$ at $(2,3)$

In

\mathrm{EvaluateForm}(\mathrm{QuadraticForm}(1, 1, -1), 2, 3)

Out

1

one class of discriminant 5

In

\mathrm{FormClassNumber}(5)

Out

1

two classes of discriminant 12

In

\mathrm{FormClassNumber}(12)

Out

2

Applications #

the class, read back as a geodesic

In

\mathrm{ModularWord}(\mathrm{FormAutomorph}(\mathrm{QuadraticForm}(1, 1, -1)))

Out

\text{LR}

Properties #

the action preserves the discriminant

In

\mathrm{FormDiscriminant}(\mathrm{FormAction}(\mathrm{QuadraticForm}(1, 1, -1), \mathrm{ModularMatrix}(1, 1, 0, 1)))

Out

5

$3^2 - 5\cdot 1^2 = 4$

In

\mathrm{PellSolution}(5)

Out

\bigl\lbrack3, 1\bigr\rbrack

$4^2 - 12\cdot 1^2 = 4$

In

\mathrm{PellSolution}(12)

Out

\bigl\lbrack4, 1\bigr\rbrack

the automorph fixes the form

In

\mathrm{FormAction}(\mathrm{QuadraticForm}(1, 1, -1), \mathrm{FormAutomorph}(\mathrm{QuadraticForm}(1, 1, -1)))

Out

\mathrm{QuadraticForm}(1, 1, -1)

the automorph is hyperbolic

In

\mathrm{ModularTrace}(\mathrm{FormAutomorph}(\mathrm{QuadraticForm}(1, 1, -1)))

Out

3

confirmed by ModularKind

In

\mathrm{ModularKind}(\mathrm{FormAutomorph}(\mathrm{QuadraticForm}(1, 1, -1)))

Out

\text{Hyperbolic}

About

Examples	15
Crosswalk	Wikipedia MathWorld Wikidata nLab

See also: ModularClasses, ContinuedFraction, RademacherSymbol