Returns whether R is maximal at the prime P, that is, if (R:O) is not contained in P, where O is the maximal order.
Given an order R and prime P of R, it returns the minimal overorders S of R with conductor (R:S) which is P-primary. The minimality assumption forces the conductor (R:S) to be exactly P. Based on [HS20].
Computes the minimal overorders of R.
Given an order R and prime P of R, it returns R and the overorders S of R with conductor (R:S) which is P-primary. We recursively produce the minimal PP-overorders where PP are primes above P. Based on [HS20].
populateoo_in_oo: BoolElt Default: false
We compute all the overorders of R.
The parameter populateoo_in_oo (default false) determines whether we should fill the attribute T`OverOrders for every overorder T of R.
The computation is based on [HS20].
> _<x> := PolynomialRing(Integers());
> f := (x^4+16)*(x^4+81);
> A := EtaleAlgebra(f);
> E := EquationOrder(A);
> oo := OverOrders(E);
> #oo;
2288
> pp := SingularPrimes(E);
> // We see that the size of the lattice of inclusions of the overorders is the product of the sizes of local component of the lattice.
> #oo eq &*[ #OverOrdersAtPrime(E,P) : P in pp ];
true
> // Now we consider only the P-overorders S for the first singular prime P.
> // We verify that there is always a positive integer i such that (R:S)^i is invertible in its multiplicator ring.
> ooP := OverOrdersAtPrime(E,pp[1]);
> #ooP;
11
> forall{exists{IsInvertible(Ti!!Ci) where Ti:=MultiplicatorRing(Ci) where Ci:=C^i : i in [1..10]} where C := ColonIdeal(E,E!!OneIdeal(S)) : S in ooP};
true
populateoo_in_oo: BoolElt Default: false
We compute all the overorders of R.
The parameter populateoo_in_oo (default false) determines whether we should fill the attribute T`OverOrders for every overorder T of R.
The computation is based on [HS20].
V2.29, 28 November 2025