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The majority of functions for quadratic fields and orders apply
identically to number fields and orders in general. The functions which
exist only for quadratic fields and orders are listed here along with
those which deserve a special mention.
Subsections
AssignNames(~O, [s]) : RngQuad, [ MonStgElt ]) ->
Procedure to change the name of the generator of a quadratic field
F or an order O in a quadratic field to the string s.
Elements of the quadratic field Q(Sqrt(d)) with
m squarefree will be printed in the form 1/b*(x + y*s),
where b, x, y are integers.
Similarly, for an order O of conductor f in a quadratic field elements
will be printed in the format x + y*s.
This procedure only changes the name used in printing the elements of F
or O, it does not make an assignment to an identifier s.
To do this, use an assignment statement,
or angle brackets when creating the field or order: F<s> := QuadraticField(-3);.
Note that since this is a procedure that modifies F or O,
it is necessary to have a reference ~ in the call to this function.
Name(O, 1) : RngQuad, RngIntElt -> RngQuadElt
Given a quadratic field F or one of its orders O, return the element
which has the name attached to it, that is, return Sqrt(d)
in the field, or f εd in a suborder of the maximal order
or f Sqrt(d) in a suborder of the equation order.
FundamentalUnit(O) : RngQuad -> RngQuadElt
A generator for the unit group of the order O or the
maximal order of the quadratic field K.
The discriminant of the field K which is only defined up to squares.
The discriminant will be the discriminant of the polynomial or better.
The conductor of the field K which is the order of the smallest cyclotomic
field containing K and a sequence containing the ramified real places of K.
The conductor of the order O, which equals the index of O
in the maximal order.
The function ClassGroup is available for number
fields and orders
in general but a different and faster algorithm is used for the quadratics.
All of the algorithms are based on binary quadratic forms,
see ClassGroup in Chapter BINARY QUADRATIC FORMS
for details.
ClassGroup(O) : RngQuad -> GrpAb, Map
FactorBasisBound: FldReElt Default: 0.1
ProofBound: FldReElt Default: 6
ExtraRelations: RngIntElt Default: 1
The class group of a maximal order O or the maximal order of the quadratic
field K, as an abelian group. The function also returns a map between
the group and the power structure of ideals of O or the maximal order
of K.
For details on the parameters see ClassGroup in Chapter
BINARY QUADRATIC FORMS.
ClassNumber(O) : RngQuad -> RngIntElt
FactorBasisBound: FldReElt Default: 0.1
ProofBound: FldReElt Default: 6
ExtraRelations: RngIntElt Default: 1
The class number of the maximal order O or the maximal order of the
quadratic field K.
PicardNumber(O) : RngQuad -> RngIntElt
FactorBasisBound: FldReElt Default: 0.1
ProofBound: FldReElt Default: 6
ExtraRelations: RngIntElt Default: 1
The picard group (the group of the invertible ideals of O modulo the
principal ones) of the order O or the size of this group.
PicardGroup also returns a map from the group to the ideals of O.
QuadraticClassGroupTwoPart(O) : RngQuad -> GrpAb, Map
QuadraticClassGroupTwoPart(d) : RngIntElt -> GrpAb, Map
Factorization: RngIntEltFact Default: []
Use the Bosma-Stevenhagen algorithm to compute the 2-part of the
class group of a quadratic order. Returned are: an array of forms
that generates the 2-part and an array that gives the orders of
the respective elements. The Factorization of the given discriminant
can be given as additional information.
> G, f := QuadraticClassGroupTwoPart(33923894057872); G;
Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4 + Z/16 + Z/16
> Random(G);
11*G.1 + 2*G.2 + G.3 + G.4 + G.6
> f($1);
<-1212992,3947508,3780131>
> G, f := QuadraticClassGroupTwoPart(QuadraticField(33923894057872)); G;
Abelian Group isomorphic to Z/2 + Z/8 + Z/16
For imaginary quadratic fields, (that is, for quadratic fields
Q(sqrt m) with m < 0), the function NormEquation is provided
specially for quadratics
to find integral elements of a given norm. For real quadratic fields
conics are used, see Section Finding Points for details.
NormEquation(F, m: parameters) : FldQuad, RngIntElt -> BoolElt, SeqEnum
NormEquation(O, m) : RngQuad, RngIntElt -> BoolElt, SeqEnum
NormEquation(O, m: parameters) : RngQuad, RngIntElt -> BoolElt, SeqEnum
Factorization: [<RngIntElt, RngIntElt>] Default:
All: BoolElt Default: true
Solutions: RngIntElt Default: All
Exact: BoolElt Default: false
Ineq: BoolElt Default: false
SetVerbose("NormEquation", n): Maximum: 1
Given quadratic field F and a non-negative integer m,
return true if there exists an element αin the ring of integers OF of
F with norm m, and false otherwise. Instead of searching
the maximal order OF it is possible to search any suborder O of OF for
such element αby supplying O as a first argument.
For imaginary quadratic fields the method used is constructive
(it uses Cornacchia's algorithm, see [Coh93] section 1.5.2),
and if the value true is returned then a solution [x] is also returned
as a second return value.
Note that if the discriminant F=Q(Sqrt(d)) with d equiv1bmod4
(and squarefree) this function searches for a solution in integers to
x2 + y2d=4m (and the solution α=(x + ySqrt(d)/2) is returned),
whereas for d = 2, 3bmod4
a solution α=x + ySqrt(d) with x2 + y2d=m in integers x, y is
returned, if it exists. In an order of conductor f a search is
conducted for a solution to the same equation with d replaced by f2d.
Note that a version of NormEquation with integer arguments d
and m also exists (see Section The Solution of Modular Equations).
Unless m is the square of an integer, the factorization of m is
used by the algorithm; if it is known, it may be supplied
as the value of the optional parameter Factorization to speed
up the calculation.
A verbose flag can be set to obtain some information on progress
with the computation (see SetVerbose).
For real quadratic fields the same algorithm is used as for the general
number fields. The last 4 parameters refer to this algorithm. See
Section Solving Norm Equations
for a description.
> d := 302401481761723680;
> m := 76814814791186002463716;
> Q<z> := QuadraticField(-d);
> O<w> := sub< MaximalOrder(Q) | 6 >;
> f, s := NormEquation(O, m);
> s, Norm(s[1]);
406 + 1008*w 76814814791186002463716
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