Return whether x is coercible into the order S and the result if so.
Check: RngIntElt Default: 100
CheckIsKnownOrder: BoolElt Default: true
Construct the order generated by gens over the rationals. The parameter Check (default 100) determines how many times the program tries to obtain a multiplicatively closed lattice by adding the product of the generators. If Check is 0 then this step is skipped. The parameter CheckIsKnownOrder determines whether the program checks if the order is already known, i.e. in the attribute KnownOrders of the algebra. This is to avoid the creation of multiple copies of the same order. The default value is true.
Given a sequence of orders in the number fields defining the étale algebra A, generates the direct sum order.
Returns the algebra of the order S.
Return a Z-basis of the order S.
Return a set of generators (as a Z-algebra) of the order S.
Checks equality of orders in an étale Algebra.
x in O : RngIntElt, AlgEtQOrd -> BoolElt
x in O : FldRatElt, AlgEtQOrd -> BoolElt
Return whether the element x is contained in the order O of an algebra.
Returns the coordinates of the elements in seq with respect to the fixed
internal basis. The basis used is returned by AbsoluteBasis.
Returns the unit element of the order S.
Returns the zero element of the order S.
ZeroDivisorsAllowed: BoolElt Default: false
Returns a random element of the order O. The coefficients are bounded by the positive integer bd. One can allow zero-divisors using the optional parameter
ZeroDivisorsAllowed, which is set to false by default.
CoeffRange: RngIntElt Default: 3
ZeroDivisorsAllowed: BoolElt Default: false
Returns a random (small coefficient) element of the order O. 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 is set to false by default.
This procedure checks whether the order R is already in the list of known orders of the algebra A containing R. If so then it replaces R with the copy stored in the attribute KnownOrders. If not it adds it to KnownOrders. This is done to avoid creating multiple copies of the same order.
Given an étale algebra defined by a polynomial, returns the monogenic order defined by the same polynomial.
Given an étale algebra A, returns the order consisting of the product of the equation orders of the number fields.
Returns the maximal order of the étale algebra A. It is the direct sum of the ring of integers of the number fields composing the algebra.
Returns whether the order S is the maximal order of the étale algebra.
Return if the order O is a product of orders in number fields, and if so return also the sequence of these orders.
Given an order T compute its index with respect to the basis of the algebra of T as a free Z-module.
Given two orders T ⊂S, returns [S:T] = #S/T.
Checks whether O1 is contained in O2.
Returns the order generated by the orders O1 and O2.
Return the intersection of orders O1 and O2.
Returns the multiplicator ring of an order R, that is R itself.
V2.29, 21 October 2025