In the lists below, K denotes a quadratic field, O a quadratic order, and B a magma of binary quadratic forms.

Subsections

• Related Structures
• Basis
• Ideal Class Group
• Unit Group
• Numerical Invariants
• Predicates and Boolean Operators

Category(K) : FldQuad -> Cat

Category(O) : RngQuad -> Cat

Category(B) : MagForm -> Cat

Parent(K) : FldQuad -> Pow

Parent(O) : RngQuad -> Pow

Parent(B) : MagForm -> Pow

PrimeField(K) : FldQuad -> FldRat

PrimeRing(O) : RngQuad -> RngInt

This function returns a sequence of elements of the quadratic field K that form an integral basis for K.

A Z-basis for the order O, as a sequence of two elements of the quadratic field K in which O is contained. The two elements are 1 and f omega , where 1, omega form the standard integral basis for K, and f is the conductor of O.

ClassGroup(O: parameters) : RngQuad -> GrpAb, Map

    Time: FldPrElt                      Default: 0.1

    Stack: FldPrElt                     Default: 0.1

    ExtraRels: RngIntElt                Default: 5

The class group of an 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 magma of quadratic forms of the associated discriminant.

ClassGroupStructure(O: parameters) : RngQuad -> [ RngIntElt ]

    Time: FldPrElt                      Default: 0.1

    Stack: FldPrElt                     Default: 0.1

    ExtraRels: RngIntElt                Default: 5

The structure of the class group of the order O or the maximal order of the quadratic field K, as a sequence of integers giving the abelian invariants.

ClassNumber(O: parameters) : RngQuad -> RngIntElt

    Al: MonStgElt                       Default: "ClassGroup"

    Time: FldPrElt                      Default: 0.1

    Stack: FldPrElt                     Default: 0.1

    ExtraRels: RngIntElt                Default: 5

The class number of O or the order O or the maximal order of the quadratic field K. The parameter Al may be supplied to select the method used to calculate the class number. The possible values are "ClassGroup" (finds the order of the class group), "ReducedForms" (finds the number of reduced forms), "Shanks" (uses the heuristic method of Shanks) or "Lseries" (uses the L-series). The default is "ClassGroup". Currently the methods "ReducedForms" and "Shanks" may only be used for imaginary quadratics fields or orders. Also, note that the heuristic method of Shanks is not guaranteed to give the correct answer, although it is usually correct. (For example, using QuadraticField(-2163679) it gives the wrong answer 760, while the other methods correctly give 800.)

UnitGroup(O) : RngQuad -> GrpAb, Map

The unit group of the order O or the maximal order of the quadratic field K, as an abelian group, together with a map to the order (or field).

TorsionSubgroup(O) : RngQuad -> GrpAb, Map

Returns the torsion part of the unit group of the order O or of the maximal order of the quadratic field K, as a finite abelian group together with a map from the group to the order O or the field K.

FundamentalUnit(O) : RngQuad -> RngQuadElt

A generator for the unit group of the order O or the maximal order of the quadratic field K.

UnitRank(O) : RngQuad -> RngIntElt

The rank of the free part of the unit group of the order O or the maximal order of the quadratic field K, which equals 1 for real quadratic fields and 0 for imagnary quadratic fields.

Characteristic(K) : FldQuad -> RngIntElt

Characteristic(O) : RngQuad -> RngIntElt

Degree(O) : RngQuad -> RngIntElt

The (absolute) degree of K over Q, or of the order O (as a Z-module), which is 2 for all quadratic fields and orders.

Discriminant(O) : FldQuad -> RngIntElt

The discriminant of the quadratic field K or of an order O of K. If K=Q(sqrt(d)), with d squarefree, this returns d if d = 0, 1 mod 4, and 4d otherwise. For the order the discriminant equals f^2 times the field discriminant, where f is the index of O in the maximal order.

The finite part of the conductor of the quadratic field K. This is the smallest positive integer n such that K is contained in Q(zeta_n). It equals the absolute value of the discriminant.

Conductor(O) : RngQuad -> RngIntElt

The conductor of order O, which equals the index of O in the ring of integers of its field of fractions.

Regulator(O) : RngQuad -> RngIntElt

The regulator of the order O or the maximal order of the quadratic field K.

The signature of the quadratic field, that is, the number of real embeddings and the number of pairs of complex embeddings of K. So this function returns either 2, 0 or 0, 1 depending on whether the field is real or imaginary quadratic.

The predicates listed below are available both for quadratic fields and for their orders.

IsCommutative(K) : FldQuad -> BoolElt

IsCommutative(O) : RngQuad -> BoolElt

IsUnitary(K) : FldQuad -> BoolElt

IsUnitary(O) : RngQuad -> BoolElt

IsFinite(K) : FldQuad -> BoolElt

IsOrdered(K) : FldQuad -> BoolElt

IsFinite(O) : RngQuad -> BoolElt

IsOrdered(O) : RngQuad -> BoolElt

IsField(K) : FldQuad -> BoolElt

IsEuclideanDomain(K) : FldQuad -> BoolElt

IsField(O) : RngQuad -> BoolElt

IsEuclideanDomain(O) : RngQuad -> BoolElt

IsPID(K) : FldQuad -> BoolElt

IsUFD(K) : FldQuad -> BoolElt

IsPID(O) : RngQuad -> BoolElt

IsUFD(O) : RngQuad -> BoolElt

IsDivisionRing(K) : FldQuad -> BoolElt

IsEuclideanRing(K) : FldQuad -> BoolElt

IsDivisionRing(O) : RngQuad -> BoolElt

IsEuclideanRing(O) : RngQuad -> BoolElt

IsPrincipalIdealRing(K) : FldQuad -> BoolElt

IsDomain(K) : FldQuad -> BoolElt

IsPrincipalIdealRing(O) : RngQuad -> BoolElt

IsDomain(O) : RngQuad -> BoolElt

K eq L : FldQuad, FldQuad -> BoolElt

O eq P : RngQuad, RngQuad -> BoolElt

K ne L : FldQuad, FldQuad -> BoolElt

O ne P : RngQuad, RngQuad -> BoolElt

True if O is the maximal order (i.e. the ring of integers) of its field of fractions, false otherwise.

True if O is the equation order of its field of fractions, false otherwise.