Ideals

Ideal(S, gens): AlgEtQOrd, SeqEnum -> AlgEtQIdl: Creates an ideal of the order \(S\), generated by the elements of \(gens\).

Ideal(S, idls): AlgEtQOrd, Tup -> AlgEtQIdl: Given an order \(S\) which is a product of orders \(S_i\) in the number fields generating Algebra(S), and a tuple of ideals \(I_i\) of \(S_i\), returns the \(S\)-ideal corresponding to the direct sum of the \(I_i\).

Ideal(S, gen): AlgEtQOrd, Any -> AlgEtQIdl
S * gen: AlgEtQOrd, AlgEtQElt -> AlgEtQIdl
S * gen: AlgEtQOrd, RngIntElt -> AlgEtQIdl
S * gen: AlgEtQOrd, FldRatElt -> AlgEtQIdl
gen * S: AlgEtQElt, AlgEtQOrd -> AlgEtQIdl
gen * S: RngIntElt, AlgEtQOrd -> AlgEtQIdl
gen * S: FldRatElt, AlgEtQOrd -> AlgEtQIdl: Creates an ideal of \(S\), generated by \(gen\).

T !! I: AlgEtQOrd, AlgEtQIdl -> AlgEtQIdl: Given an \(S\)-ideal \(I\) and an order \(T\), returns the extension \(IT\) as a \(T\)-ideal. Note that if \(T\) is a subset of \(S\), then \(IT=I\).

Algebra(I): AlgEtQIdl -> AlgEtQ: Returns the étale algebra in which the ideal \(I\) lives.

Order(I): AlgEtQIdl -> AlgEtQOrd: Returns the order of definition of the ideal \(I\).

ZBasis(I): AlgEtQIdl -> SeqEnum[AlgEtQElt]: Returns a \({{\Z}}\)-basis of the ideal \(I\).

Generators(I): AlgEtQIdl -> SeqEnum[AlgEtQElt]: Returns the generators of the ideal \(I\).

I eq J: AlgEtQIdl, AlgEtQIdl -> BoolElt

I ne J: AlgEtQIdl, AlgEtQIdl -> BoolElt

Returns whether the ideals \(I\) and \(J\) are equal, respectively not equal.

Let \(I\) a fractional \(S\)-ideal in an étale algebra over \({{\Q}}\). Let \(P\) be the upper triangular Hermite normal form of the integer square matrix \(d\cdot M\) where \(M\) is the matrix whose rows are the coefficients of a \({{\Z}}\)-basis of \(I\) and \(d\) is the least common denominator of its entries. The Hash of \(I\) is defined to be the sequence consisting of the least common denominator of \({1\over d}\cdot P\) and the entries of the upper triangular part of \({1\over d}\cdot P\). This hashing method has no collisions and it is independent of the choice of \({{\Z}}\)-basis from which we start the procedure.

I eq S: AlgEtQIdl, AlgEtQOrd -> BoolElt
S eq I: AlgEtQOrd, AlgEtQIdl -> BoolElt: Return whether \(I\) is equal to \(S\) when \(I\) is an ideal of \(S\).

AbsoluteCoordinates(x, I): AlgEtQElt, AlgEtQIdl -> SeqEnum: Given an element \(x\) and an ideal \(I\), returns the coordinates of \(x\) with respect to the stored Z-basis of \(I\).

AbsoluteCoordinates(seq, I): SeqEnum[AlgEtQElt], AlgEtQIdl -> SeqEnum: Returns the coordinates of the elements in seq with respect to the stored Z-basis of \(I\).

x in I: AlgEtQElt, AlgEtQIdl -> BoolElt
x in I: RngIntElt, AlgEtQIdl -> BoolElt
x in I: FldRatElt, AlgEtQIdl -> BoolElt: Returns whether the element \(x\) is in the ideal \(I\).

S subset I: AlgEtQOrd, AlgEtQIdl -> BoolElt: Given an ideal \(I\) of an order \(S\), return whether \(S \subseteq I\).

I subset S: AlgEtQIdl, AlgEtQOrd -> BoolElt: Given an ideal \(I\) of an order \(S\), return whether \(I \subseteq S\).

I1 subset I2: AlgEtQIdl, AlgEtQIdl -> BoolElt: Checks if the ideal \(I1\) is inside the ideal \(I2\). The ideals need to be fractional.

Index(T): AlgEtQIdl -> FldRatElt: Given an ideal \(T\) computes its index with respect to the basis of the algebra of \(T\) as a free \({{\Q}}\)-module.

Index(J, I): AlgEtQIdl, AlgEtQIdl -> Any: Given fractional ideals \(J\) and \(I\) defined over the same order returns \([J:I] = [J:J \cap I]/[I : J \cap I]\).

Index(S, I): AlgEtQOrd, AlgEtQIdl -> Any: Given an ideal \(I\) of an order \(S\) returns \([S:I] = [S:S \cap I]/[I : S \cap I]\).

OneIdeal(S): AlgEtQOrd -> AlgEtQIdl: Given an order \(S\) returns the ideal \(1*S\) which will be cached.

Conductor(O): AlgEtQOrd -> AlgEtQOrdIdl: Computes the conductor of an order \(O\), defined as the colon ideal \((O:O_K)\), where \(O_K\) is the maximal order of the algebra.

I + J: AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl: Returns the sum of two ideals.

I * J: AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl: Product of two ideals.

I * x: AlgEtQIdl, AlgEtQElt -> AlgEtQIdl
I * x: AlgEtQIdl, RngIntElt -> AlgEtQIdl
I * x: AlgEtQIdl, FldRatElt -> AlgEtQIdl
x * I: AlgEtQElt, AlgEtQIdl -> AlgEtQIdl
x * I: RngIntElt, AlgEtQIdl -> AlgEtQIdl
x * I: FldRatElt, AlgEtQIdl -> AlgEtQIdl: Returns \(x*I\).

I ^ n: AlgEtQIdl, RngIntElt -> AlgEtQIdl: Returns the \(n\)th power of an ideal.

I meet S: AlgEtQIdl, AlgEtQOrd -> AlgEtQIdl
S meet I: AlgEtQOrd, AlgEtQIdl -> AlgEtQIdl: Given an ideal \(I\) of \(S\), return \(S \cap I\).

I meet J: AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl: Given ideals \(I\) and \(J\), return \(J \cap I\).

&+ seq: SeqEnum[AlgEtQIdl] -> AlgEtQIdl: Returns the sum of the fractional ideals in the sequence.

ColonIdeal(I, J): AlgEtQIdl, AlgEtQIdl -> AlgEtQIdl: Computes the colon ideal \((I:J)\) (as an \(O\)-ideal) of two \(O\)-ideals, which is the set of elements \(x\) of the algebra such that \(x \cdot J \subset I\).

ColonIdeal(O, J): AlgEtQOrd, AlgEtQIdl -> AlgEtQIdl: Computes the colon ideal \((1 \cdot O:J)\) (as an \(O\)-ideal).

ColonIdeal(I, O): AlgEtQIdl, AlgEtQOrd -> AlgEtQIdl: Computes the colon ideal \((I:1 \cdot O)\) (as an \(O\)-ideal).

IsInvertible(I): AlgEtQIdl -> BoolElt: Checks if the ideal \(I\) is invertible in its order of definition \(O\).

Inverse(I): AlgEtQIdl -> AlgEtQIdl: Computes the inverse of an invertible ideal \(I\).

MultiplicatorRing(I): AlgEtQIdl -> AlgEtQOrd: Given a fractional ideal \(I\) computes its multiplicator ring \((I:I)\).

IsProductOfIdeals(I): AlgEtQIdl -> BoolElt, Tup: Return if the ideal \(I\) is a product of ideals in the number fields defining the algebra. If so, it returns also the sequence of these ideals (in the appropriate orders). Note: we require Order(I) to be MultiplicatorRing(I).

Random(I, bd): AlgEtQIdl, RngIntElt -> AlgEtQElt

ZeroDivisorsAllowed: BoolElt                    Default: false

Returns a random element of the ideal \(I\). The coefficients are bounded by the positive integer \(bd\). One can allow zero-divisors using the optional parameter ZeroDivisorsAllowed, which by default is set to false.

Random(I): AlgEtQIdl -> AlgEtQElt

CoeffRange         : RngIntElt                    Default: 3
ZeroDivisorsAllowed: BoolElt                      Default: false

Returns a random (small coefficient) element of the ideal \(I\). The range of the random coefficients can be increased by giving the optional parameter CoeffRange. One can allow zero-divisors using the optional parameter ZeroDivisorsAllowed, which by default is set to false.

IsCoprime(I, J): AlgEtQIdl, AlgEtQIdl -> BoolElt: Given two integral ideals \(I\) and \(J\) of an order \(S\), returns whether \(I+J=R\).

IsIntegral(I): AlgEtQIdl -> BoolElt: Returns whether the ideal \(I\) of \(S\) is integral, that is \(I \subseteq S\).

MakeIntegral(I): AlgEtQIdl -> AlgEtQIdl, RngIntElt: Given a fractional \(S\) ideal \(I\), returns the ideal \(d \cdot I, d\) when \(d\) is the smallest integer such that \(d \cdot I\) is integral in \(S\). Compare with SmallRepresentative.

MinimalInteger(I): AlgEtQIdl -> RngIntElt: Returns the smallest integer contained in the ideal \(I\).

CoprimeRepresentative(I, J): AlgEtQIdl, AlgEtQIdl -> AlgEtQElt, AlgEtQIdl: Returns an element \(x\) such that \(x \cdot I\) is an integral ideal coprime with \(J\), together with the product \(x \cdot I\). The first ideal must be invertible and the second should be integral.

ZBasisLLL(~S): AlgEtQOrd
ZBasisLLL(~S): AlgEtQIdl: A procedure that replaces the ZBasis with an LLL-reduced one. Note: the attribute inclusion matrix, which depends on the \({{\Z}}\)-Basis is modified as well.

Ideals#

Ideals