Basic Invariants

Structures of binary quadratic forms are defined in terms of a discriminant, and membership in a structure determined by this invariant. To aid in the construction of forms, additional elementary functions are provided to test integer inputs to determine if they define valid discriminants of quadratic forms.

Discriminant(f): QuadBinElt -> RngIntElt

The discriminant \(b^2-4ac\) of a quadratic form \(f=aX^2+bXY+cY^2\).

Discriminant(Q): QuadBin -> RngIntElt

The discriminant of the quadratic forms in the structure \(Q\).

IsDiscriminant(D): RngIntElt -> BoolElt

True iff the integer \(D\) is the discriminant of some quadratic form. This holds iff \(D\) is congruent to \(0\) or \(1\) mod 4.

IsFundamental(D): RngIntElt -> BoolElt

IsFundamentalDiscriminant(D): RngIntElt -> BoolElt

True iff the integer \(D\) is a fundamental discriminant, i.e. a discriminant that is not a square multiple of any smaller discriminant.

FundamentalDiscriminant(D): RngIntElt -> RngIntElt

The fundamental discriminant \(D_0\) such that \(D\) is a square multiple of \(D_0\).

Conductor(Q): QuadBin -> RngIntElt

Let \(D\) be Discriminant(Q). The conductor is the integer \(m\) such that \(D = m^2 D_0\), where \(D_0\) is the fundamental discriminant.