Given a form f = ax2 + bxy + cy2, returns the conjugate form
ax2 - bxy + cy2.
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.
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.
The right action of SL(2, Z) on the set of quadratic forms of discriminant D
is given by the rule
f(x, y) ((r s atop t u)) = f(rx + sy, tx + uy).
The right action of M ∈SL(2, Z) on the binary quadratic form f.
ReducedForm(f) : QuadBinElt -> QuadBinElt, Mtrx
Returns a reduced quadratic form equivalent to f,
and the transformation matrix.
The result of applying one reduction step to the quadratic form f.
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).
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).
The coefficient sequence can be accessed as a sequence of integers,
providing the inverse operation to the forms coercion constructor.
The i-th coefficient of f, where 1 ≤i ≤3.
ElementToSequence(f) : QuadBinElt -> SeqEnum[RngIntElt]
The sequence [a, b, c] where f is the form ax2 + bxy + cy2.
Return true if and only if f is in Q, that is f and Q have the same
discriminant.
Return true if the quadratic form f and g are equal and false otherwise.
Return true if and only if f is the principal form in its parent structure.
Return true if the quadratic form f is reduced; false otherwise.
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 < a, b, c > and < - a, b, - c > 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.
The quotient homomorphism from the class group of Q to the class group
of fundamental discriminant.
Given two structures of quadratic forms Q1 and Q2, such that the
discriminant of Q1 equals a square times the discriminant of Q2,
the quotient homomorphism from Q1 to Q2 is returned as a map.
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 m2D for some integer m.
Parent(f) : QuadBinElt -> QuadBin
Category(Q) : QuadBin -> Cat
Given a structure of quadratic forms of discriminant D,
returns the quadratic order of discriminant D.
Given a quadratic form f = ax2 + bxy + cy2, returns the ideal
(a, ( - b + Sqrt(D))/2) in the quadratic order Z[(t + Sqrt(D))/2],
where t equals 0 or 1.
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