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The reduced primitive forms of discriminant $- 8$

Example

The only reduced primitive positive-definite binary quadratic form of discriminant $- 8$ is

$x^{2} + 2 y^{2} = (1, 0, 2) .$

Consequently $h (- 8) = 1$ , and every primitive positive-definite form of discriminant $- 8$ is properly equivalent to $x^{2} + 2 y^{2}$ .

Facts & Assumptions

Given: A reduced primitive positive-definite form $(a, b, c)$ of discriminant $- 8$ .

[L1]

A reduced form of discriminant $Δ$ satisfies $a \leq \sqrt{∣ Δ ∣ / 3}$ (A reduced positive-definite form of discriminant $Δ$ satisfies $a \leq \sqrt{∣ Δ ∣ / 3}$ ).

[L2]

Each proper-equivalence class contains exactly one reduced form (Each proper-equivalence class of positive-definite integral binary quadratic forms contains exactly one reduced form).

[L3]

The class number $h (Δ)$ counts proper-equivalence classes of primitive positive-definite forms of discriminant $Δ$ (The class number of primitive positive-definite binary quadratic forms of discriminant $Δ$ ).

Verification

technique · direct

1.1

Here $a \leq \sqrt{8 / 3} < 2$ , so the positive integer $a$ must be $1$ .

L1givenalgebra

2.1

The discriminant equation gives $- 8 = b^{2} - 4 c$ , so $4 c = b^{2} + 8$ . Reducedness gives $∣ b ∣ \leq 1$ ; the choices $b = \pm 1$ make $c = 9 / 4$ , not an integer, while $b = 0$ gives $c = 2$ . Thus the only reduced possibility is $(1, 0, 2)$ .

step 1.1algebra

3.1

The form $(1, 0, 2)$ is primitive, so by [L2] there is exactly one proper-equivalence class of primitive positive-definite forms of discriminant $- 8$ . Hence [L3] gives $h (- 8) = 1$ , and every such form is properly equivalent to $(1, 0, 2)$ .

L2L3step 2.1∎

Depends on

Used by

Odd primes congruent to 1 or 3 modulo 8 are represented by x²+2y² Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

William Stein, Elementary Number Theory and Elliptic Curves, Section 9.4 (standard reference, not scraped)

The reduced primitive forms of discriminant −8−8