Operations on Forms

Arithmetic

Conjugate(f): QuadBinElt -> QuadBinElt

Given a form \(f = ax^2 + bxy + cy^2\), returns the conjugate form \(ax^2 - bxy + cy^2\).

f * g: QuadBinElt, QuadBinElt -> QuadBinElt
Composition(f, g): QuadBinElt, QuadBinElt -> QuadBinElt
Al       : MonStgElt                    Default: "Gauss"
Reduction: BoolElt                      Default: false

Returns the composition of two binary quadratic forms \(f\) and \(g\). The operator `*' returns a reduced form equivalent to the product using a fast composition algorithm of Shanks. In contrast, by default Composition returns the true composition in the group of forms, unless the parameter Reduction := true. The function Composition takes a further parameter Al which specifies whether the algorithm of Gauss or Shanks, set to "Gauss" by default. The algorithm of Shanks performs partial intermediate reductions, so it is not allowed to select Al := "Shanks" and Reduction := false.

f ^ n: QuadBinElt, RngIntElt -> QuadBinElt
Power(f, n): QuadBinElt, RngIntElt -> QuadBinElt
Al       : MonStgElt                    Default: "Gauss"
Reduction: BoolElt                      Default: false

Returns the \(n\)-th power of a form \(f\). The operator `^' returns a reduced representative, using the fast composition algorithm of Shanks. In contrast, by default Power returns the true composition in the group of forms, unless the parameter Reduction := true. The function Power takes the further parameter Al which specifies whether the algorithm of Gauss or Shanks is used, set to "Gauss" by default. The algorithm of Shanks performs partial intermediate reductions, so it is not allowed to select Al := "Shanks" and Reduction := false.

Matrix Action

The right action of \({\operatorname{SL}}(2,{\mathbb{Z}})\) on the set of quadratic forms of discriminant \(D\) is given by the rule

\[f(x,y) \left({r\ s\atop t\ u}\right) = f(rx+sy,tx+uy).\]
f * M: QuadBinElt, GrpMatElt -> QuadBinElt

The right action of \(M \in {\operatorname{SL}}(2,{\mathbb{Z}})\) on the binary quadratic form \(f\).

Reduction

Reduction(f): QuadBinElt -> QuadBinElt, Mtrx
ReducedForm(f): QuadBinElt -> QuadBinElt, Mtrx

Returns a reduced quadratic form equivalent to \(f\), and the transformation matrix.

ReductionStep(f): QuadBinElt -> QuadBinElt

The result of applying one reduction step to the quadratic form \(f\).

ReductionOrbit(f): QuadBinElt -> {@ QuadBinElt @}

For a binary quadratic form \(f\) of positive discriminant, this returns an indexed set containing all reduced forms equivalent to \(f\). These are obtained by iterating the ReductionStep operator, starting with Reduction(f).

Order(f): QuadBinElt -> RngIntElt

For a binary quadratic form \(f\) of discriminant \(D\), this returns the order of \(f\) in the class group for discriminant \(D\) (see ClassGroup). Note that the class group is defined using the equivalence relation IsEquivalent(f1, f2 : Narrow := false).

Attribute Access

The coefficient sequence can be accessed as a sequence of integers, providing the inverse operation to the forms coercion constructor.

f[i]: QuadBinElt, RngIntElt -> RngIntElt

The \(i\)-th coefficient of \(f\), where \(1 \le i \le 3\).

Eltseq(f): QuadBinElt -> SeqEnum[RngIntElt]
ElementToSequence(f): QuadBinElt -> SeqEnum[RngIntElt]

The sequence \([a,b,c]\) where \(f\) is the form \(ax^2 + bxy + cy^2\).

Boolean Operations

f in Q: QuadBinElt, QuadBin -> BoolElt

Return true if and only if \(f\) is in \(Q\), that is \(f\) and \(Q\) have the same discriminant.

f eq g: QuadBinElt, QuadBinElt -> BoolElt

Return true if the quadratic form \(f\) and \(g\) are equal and false otherwise.

IsIdentity(f): QuadBinElt -> BoolElt

Return true if and only if \(f\) is the principal form in its parent structure.

IsReduced(f): QuadBinElt -> BoolElt

Return true if the quadratic form \(f\) is reduced; false otherwise.

IsEquivalent(f, g): QuadBinElt, QuadBinElt -> BoolElt, AlgMatElt
Narrow: BoolElt                    Default: true

For binary quadratic forms \(f\) and \(g\) of the same discriminant, this returns whether the forms are equivalent; that is, whether they can be obtained from each other by a unimodular transformation. When the forms are equivalent, a transformation matrix is also returned.

If the parameter Narrow is false, the function tests a different notion of equivalence in the case of positive discriminant, which may be weaker. This is defined by merging the equivalence classes of forms \(\langle a,b,c \rangle\) and \(\langle -a,b,-c \rangle\) into a single equivalence class. (Another way to describe it is as follows: Forms are equivalent iff their associated ideals are in the same ordinary ideal class. On the other hand, “narrow” equivalence of forms corresponds to narrow ideal classes.) In this case, the transformation matrix may produce the complementary form of \(g\).

Maps of Forms

FundamentalQuotient(Q): QuadBin -> Map

The quotient homomorphism from the class group of \(Q\) to the class group of fundamental discriminant.

QuotientMap(Q1, Q2): QuadBin, QuadBin -> Map

Given two structures of quadratic forms \(Q_1\) and \(Q_2\), such that the discriminant of \(Q_1\) equals a square times the discriminant of \(Q_2\), the quotient homomorphism from \(Q_1\) to \(Q_2\) is returned as a map.

Q ! f: QuadBin, QuadBinElt -> QuadBinElt

The ! operator coerces a binary quadratic form \(f\) into the structure \(Q\), when they are compatible. This requires that \(Q\) is QuadraticForms(D) and \(f\) has discriminant \(m^2D\) for some integer \(m\).