Creates an ideal of the order S, generated by the elements of gens.
Given an order S which is a product of orders Si in the number fields generating Algebra(S), and a tuple of ideals Ii of Si, returns the S-ideal corresponding to the direct sum of the Ii.
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.
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.
Returns the étale algebra in which the ideal I lives.
Returns the order of definition of the ideal I.
Returns a Z-basis of the ideal I.
Returns the generators of the ideal I.
I ne J : AlgEtQIdl, AlgEtQIdl -> BoolElt
Returns whether the ideals I and J are equal, respectively not equal.
S eq I : AlgEtQOrd, AlgEtQIdl -> BoolElt
Return whether I is equal to S when I is an ideal of S.
Returns the coordinates of the elements in seq with respect to the
internal fixed basis. The basis used is returned by AbsoluteBasis.
x in I : RngIntElt, AlgEtQIdl -> BoolElt
x in I : FldRatElt, AlgEtQIdl -> BoolElt
Returns whether the element x is in the ideal I.
Given an ideal I of an order S, return whether S ⊆I.
Given an ideal I of an order S, return whether I ⊆S.
Checks if the ideal I1 is inside the ideal I2. The ideals need to be fractional.
Given an ideal T computes its index with respect to the basis of the algebra of T as a free Q-module.
Given fractional ideals J and I defined over the same order returns
[J:I] = [J:J ∩I]/[I : J ∩I].
Given an ideal I of an order S returns [S:I] = [S:S ∩I]/[I : S ∩I].
Given an order S returns the ideal 1 * S which will be cached.
Computes the conductor of an order O, defined as the colon ideal (O:OK), where OK is the maximal order of the algebra.
Returns the sum of two ideals.
Product of two ideals.
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.
Returns the nth power of an ideal.
S meet I : AlgEtQOrd, AlgEtQIdl -> AlgEtQIdl
Given an ideal I of S, return S ∩I.
Given ideals I and J, return J ∩I.
Returns the sum of the fractional ideals in the sequence.
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 .J ⊂I.
Computes the colon ideal (1 .O:J) (as an O-ideal).
Computes the colon ideal (I:1 .O) (as an O-ideal).
Checks if the ideal I is invertible in its order of definition O.
Computes the inverse of an invertible ideal I.
Given a fractional ideal I computes its multiplicator ring (I:I).
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).
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.
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.
Given two integral ideals I and J of an order S, returns whether I + J=R.
Returns whether the ideal I of S is integral, that is I ⊆S.
Given a fractional S ideal I, returns the ideal d .I,
d when d is the smallest integer such that d .I is integral in S.
Compare with SmallRepresentative.
Returns the smallest integer contained in the ideal I.
Returns an element x such that x .I is an integral ideal coprime with J,
togheter with the product x .I.
The first ideal must be invertible and the second should be integral.
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.
V2.29, 21 October 2025