Factorization and Primes

Factorization(I) : AlgEtQIdl -> Tup
Given an integral S-ideal I coprime with the conductor of S (hence invertible in S), returns its factorization into a product of primes of S.
PrimesAbove(I) : AlgEtQIdl -> SeqEnum[AlgAssEtOrdIdl]
Given an integral S-ideal I, returns the sequence of maximal ideals P of S above I.
SingularPrimes(R) : AlgEtQOrd -> SeqEnum[AlgAssEtOrdIdl]
Returns the non-invertible primes of the order R.
NonInvertiblePrimes(R) : AlgEtQOrd -> SetIndx
Returns the non-invertible primes of the order R.
IsPrime(I) : AlgEtQIdl -> BoolElt
Given an integral S-ideal I, returns if the ideal is a prime fractional ideal of S, that is a maximal S ideal.
IsBassAtPrime(S, P) : AlgEtQOrd, AlgEtQIdl -> BoolElt
Check if the order is Bass at the prime ideal P, that is, if every overorder of S is Gorenstein at the primes above P.
IsBass(S) : AlgEtQOrd -> BoolElt
Check if the order S is Bass, that is, if every overorder of S is Gorenstein.
IsGorensteinAtPrime(S, P) : AlgEtQOrd, AlgEtQIdl -> BoolElt
Check if the order S is Gorenstein at the prime ideal P, that is, if every fractional ideal I with (I:I)=S is locally principal at P.
IsGorenstein(O) : AlgEtQOrd->BoolElt
Checks if the order O is Gorenstein, that is if the TraceDualIdeal of O is invertible, or equivalently, if all fractional ideals I with (I:I)=O are invertible.
V2.29, 21 October 2025