‣ LenstraBase ( n, stabilizer, super, m ) | ( function ) |
Let n and m be positive integers such that m divides n, stabilizer be a list of prime residues modulo n, which describes a subfield of the n-th cyclotomic field (see GaloisStabilizer
(60.2-5)), and super be a list representing a supergroup of the group given by stabilizer.
LenstraBase
returns a list of lists, each consisting of integers such that the elements E(n)
form a basis of the abelian number field NF( n, stabilizer )
, as a vector space over the m-th cyclotomic field (see AbelianNumberField
(60.1-2)).
This basis is an integral basis, that is, exactly the integral elements in NF( n, stabilizer )
have integral coefficients. (For details about this basis, see [Bre97].)
If possible then the result is chosen such that the group described by super acts on it, consistently with the action of stabilizer, i.e., each orbit of super is a union of orbits of stabilizer. (A usual case is super =
stabilizer, so there is no additional condition.
Note: The are in general not sets, since for stabilizer = super
, the first entry is always an element of ZumbroichBase( n, m )
; this property is used by NF
(60.1-2) and Coefficients
(61.6-3) (see 60.3).
stabilizer must not contain the stabilizer of a proper cyclotomic subfield of the n-th cyclotomic field, i.e., the result must describe a basis for a field with conductor n.
gap> LenstraBase( 24, [ 1, 19 ], [ 1, 19 ], 1 ); [ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ] gap> LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ], 1 ); [ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ] gap> LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ), 1 ); [ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]
The first two results describe two bases of the field , the third result describes a normal basis of .