Complex Multiplication

CMType(seq) : SeqEnum[Map] -> AlgEtQCMType
Given a sequence seq of homomorphisms from a CM-algebra to CC, one per conjugate pair, it returns the corresponding CMType.
CreateCMType(seq) : SeqEnum[Map] -> AlgEtQCMType
Given a sequence seq of homomorphisms from a CM-algebra to CC, one per conjugate pair, it returns the corresponding CMType.
CMType(b) : AlgEtQElt -> AlgEtQCMType
Given a totally imginary element b, it returns the CMType PHI for which b is PHI-positive, that is, (Im)(φ(b))>0 for every φ in PHI.
CreateCMType(b) : AlgEtQElt -> AlgEtQCMType
Given a totally imginary element b, it returns the CMType PHI for which b is PHI-positive.
CMPositiveElement(PHI) : AlgEtQCMType ->AlgEtQElt
Given a CMType PHI returns a totally imaginary PHI-positive element (which uniquely determines PHI).
CMPosElt(PHI) : AlgEtQCMType ->AlgEtQElt
Given a CMType PHI returns a totally imaginary PHI-positive element (which uniquely determines PHI).
Homs(PHI) : AlgEtQCMType ->SeqEnum[Map]
    prec: RngIntElt                     Default: 30
Given an AlgEtQCMType PHI returns the sequence of maps to the complex field. The parameter prec (default value 30) determines the precision of the codomains of the maps.
PHI1 eq PHI2 : AlgEtQCMType, AlgEtQCMType ->BoolElt
    prec: RngIntElt                     Default: 30
Returns whether two cm types are equal. This happens if and only if the quotient of (any) two CMPositiveElements is totally real and totally positive.
Precision(PHI) : AlgEtQCMType->RngIntElt
Returns the precision of the given CM-type, that is, the codomain of each homomorphism will be ComplexField(Precision).
ChangePrecision(PHI0, prec) : AlgEtQCMType, RngIntElt ->AlgEtQCMType
Changes the precision of the given CM-type, that is, the codomain of each homomorphism will be ComplexField(Precision).
ChangePrecision(~PHI, prec) : AlgEtQCMType, RngIntElt ->
Changes the precision of the given CM-type, that is, the codomain of each homomorphism will be ComplexField(Precision).
AllCMTypes(A) : AlgEtQ -> SeqEnum[AlgEtQCMType]
    Precision: RngIntElt                Default: 30
Returns all the AlgEtQCMTypes of A. The parameter Precision determined the precision of the codomain of the maps defining the CMTypes.
V2.29, 21 October 2025