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The class number of primitive positive-definite binary quadratic forms of discriminant $Δ$

Definition

Let $Δ < 0$ be an integer with $Δ \equiv 0$ or $1 (m o d 4)$ . The binary quadratic form class number $h (Δ)$ is the number of proper-equivalence classes of primitive positive-definite integral binary quadratic forms of discriminant $Δ$ .

This number is finite by For each negative discriminant, there are only finitely many proper-equivalence classes of positive-definite integral binary quadratic forms, since primitive positive-definite forms of discriminant $Δ$ are a subclass of all positive-definite forms of discriminant $Δ$ .

Remarks

The adjective "form" matters: later pages may compare this quantity with class numbers defined through ideals in quadratic orders.
The companion page computes $h (- 4)$ , $h (- 8)$ , $h (- 20)$ , and $h (- 23)$ by enumerating reduced representatives.

Depends on

Used by

The reduced primitive forms of discriminant -20 Example
The reduced primitive forms of discriminant -23 Example
The reduced primitive forms of discriminant -4 Example
The reduced primitive forms of discriminant -8 Example

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Sources

William Stein, Elementary Number Theory and Elliptic Curves, Definition 9.4.3 (standard reference, not scraped)

The class number of primitive positive-definite binary quadratic forms of discriminant ΔΔ