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The reduced primitive forms of discriminant $- 23$

Example

The reduced primitive positive-definite binary quadratic forms of discriminant $- 23$ are

(1, 1, 6), (2, 1, 3), (2, - 1, 3) .

Consequently $h (- 23) = 3$ .

Facts & Assumptions

Given: A reduced primitive positive-definite form $(a, b, c)$ of discriminant $- 23$ .

[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 $a \leq \sqrt{23 / 3} < 3$ , so $a$ is $1$ or $2$ .

L1givenalgebra

2.1

If $a = 1$ , then $- 23 = b^{2} - 4 c$ , so $4 c = b^{2} + 23$ . Reducedness gives $∣ b ∣ \leq 1$ . The value $b = 0$ gives no integer $c$ , while $b = \pm 1$ gives $c = 6$ ; the boundary rule forces $b = 1$ . Thus $(1, 1, 6)$ is the only reduced possibility with $a = 1$ .

step 1.1algebra

2.2

If $a = 2$ , then $- 23 = b^{2} - 8 c$ , so $8 c = b^{2} + 23$ . Reducedness gives $∣ b ∣ \leq 2$ . The values $b = 0, \pm 2$ give no integer $c$ , while $b = \pm 1$ gives $c = 3$ , and both signs are allowed because $∣ b ∣ < a$ . Thus the reduced possibilities are $(2, 1, 3)$ and $(2, - 1, 3)$ .

step 1.1algebra

3.1

The three forms from steps 2.1 and 2.2 are primitive and distinct, and [L2] shows that no other reduced primitive form of discriminant $- 23$ exists. Hence [L3] gives $h (- 23) = 3$ .

L2L3step 2.1step 2.2∎

Depends on

Used by

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Dependency tree · two levels

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Sources

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

The reduced primitive forms of discriminant −23−23