Subsections

• Creation of Structures
• Creation of Elements
• Creation of Forms

Squarefree integers determine quadratic fields. Associated with any quadratic field is its ring of integers (maximal order), and for every positive integer f there exists an order of conductor f inside the maximal order.

For any fixed discriminant the common parent of all binary quadratic forms of discriminant d can be created.

QuadraticField(m) : RngIntElt -> FldQuad

Given an integer m that is not a square, create the field Q(sqrt(d)), where d is the squarefree part of m. It is possible to assign a name to sqrt(d) using angle brackets: R<s> := QuadraticField(m).

Creation of the order Z[sqrt(d)] in the quadratic field F=Q(sqrt(d)), with d squarefree.

IntegerRing(F) : FldQuad -> RngQuad

Given a quadratic field F=Q(sqrt(d)), with d squarefree, create its maximal order. This order is Z[sqrt(d)] if d = 2, 3 mod 4 and Z[(1 + sqrt(d)/2)] if d equiv1bmod4.

FieldOfFractions(F) : FldQuad -> FldQuad

Given a quadratic order, this returns the quadratic field of which it is an order.

Create the sub-order of conductor f in the maximal order O of a quadratic field.

Create the magma of binary quadratic forms of discriminant D, with coefficients in Z.

One(K) : FldQuad -> FldQuadElt

Identity(K) : FldQuad -> FldQuadElt

Zero(K) : FldQuad -> FldQuadElt

One(O) : RngQuad -> RngQuadElt

Identity(O) : RngQuad -> RngQuadElt

Zero(O) : RngQuad -> RngQuadElt

Representative(K) : FldQuad -> FldQuadElt

Representative(O) : RngQuad -> RngQuadElt

elt< F | a_0, a_1> : FldQuad, FldRatElt , FldRatElt -> FldQuadElt

Given the quadratic field F=Q(sqrt(d)) and rational numbers a_0, a_(1), construct the element a_0 + a_1 sqrt(d) of F. Automatic coercion certifies that a_0 and a_1 are allowed to be integers.

O ! [a_0, a_1] : RngQuad, [RngIntElt] -> RngQuadElt

elt< O | a_0, a_1> : RngQuad, RngIntElt , RngIntElt -> RngQuadElt

Given an order O and integers a_0, a_1, construct the element a_0 + a_1 f epsilon_d of O, where f is the conductor of O.

F ! a : FldQuad, RngQuadElt -> FldQuadElt

O ! a : RngQuad, RngQuadElt -> RngQuadElt

Coerce a into the quadratic field F or the order O. For coercion into an order the element a must be an integer or an element of a suborder of O; for coercion into a quadratic field F a is allowed to be an integer, a rational number, or an element of F or one of its suborders.

ElementToSequence(a) : FldQuadElt -> [ FldRatElt ]

ElementToSequence(a) : RngQuadElt -> [ RngIntElt ]

Eltseq(a) : RngQuadElt -> [ RngIntElt ]

For an element a of the quadratic field Q(sqrt(d)), this returns the sequence [x, y] of rational coefficients, so that a=x + y sqrt(d). For an element a of a quadratic order O, it returns the sequence [x, y] of integers such that a=x + f epsilon_d y, where f is the conductor of O.

> Q<z> := QuadraticField(5);         
> Q eq QuadraticField(45);    
true
> O<w> := sub< MaximalOrder(Q) | 7 >;
> O;
Order of conductor 7 in Quadratic Field Q(sqrt5)
> w;
w
> Q ! w;                    
1/2*(7 + 7*z)
> Eltseq(w), Eltseq(Q ! w);
[ 0, 1 ]
[ 7/2, 7/2 ]
> ( (7/2)+(7/2)*z )^2;
1/2*(147 + 49*z)
>  Q ! w^2;
1/2*(147 + 49*z)
> w^2;
49 + 7*w

Binary quadratic forms may be created by coercing a triple of integer coefficient into the magma of forms with the right discriminant.

One(B) : MagForm -> MagFormElt

B ! 1 : MagForm, RngIntElt -> MagFormElt

MinusOne(B) : MagForm -> MagFormElt

B ! -1 : MagForm, RngIntElt -> MagFormElt

Create the principal form or its inverse, in the magma B of binary quadratic forms of discriminant D. Note that the principal form is either X^2 - D/4Y^2 or X^2 + XY + (D - 1)/4Y^2, depending on D being 0bmod4 or 1bmod4.

B ! <a, b, c> : MagForm, RngIntElt, RngIntElt, RngIntElt -> MagFormElt

Create the binary quadratic form aX^2 + bXY + cY^2 in the magma of forms B of discriminant D. An error occurs if b^2 - 4ac is not equal to D.

elt< B | a, b> : MagForm, RngIntElt, RngIntElt -> MagFormElt

Create the binary quadratic form aX^2 + bXY + cY^2 in the magma of forms B of discriminant D. Here c is determined by D=b^2 - 4ac; if no integer c exists satisfying this, an error will occur.

Create the quadratic form pX^2 + bXY + cY^2, for prime p, in the magma B of binary quadratic forms if it exists.

Return the sequence of all reduced quadratic forms for the negative fundamental discriminant D.