DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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The principal binary quadratic form of a discriminant

Definition

Let $Δ$ be an integer with $Δ \equiv 0$ or $1 (m o d 4)$ (An integer is the discriminant of an integral binary quadratic form exactly when it is congruent to $0$ or $1$ modulo $4$ ).

The principal binary quadratic form of discriminant $Δ$ is

${\begin{cases} (1, 0, - Δ / 4), & if Δ \equiv 0 (m o d 4), \\ (1, 1, (1 - Δ) / 4), & if Δ \equiv 1 (m o d 4) . \end{cases}$

By the preceding proposition, both coefficient triples are integral in their respective cases, and by direct calculation each has discriminant $Δ$ .

Remarks

This is terminology only: no class-group structure is asserted here.
Later examples identify the principal form explicitly at small discriminants, such as $x^{2} + y^{2}$ at $Δ = - 4$ and $x^{2} + 2 y^{2}$ at $Δ = - 8$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, Definition 9.2.10 (standard reference, not scraped)