# 15.10 Integral Bases for Number Fields

LenstraBase( n, stabilizer, super )

returns a list '[ ' b_1, b_2, ldots, b_m ' ]' of lists, each b_i consisting of integers such that the elements sum_{j in b_i} 'E(n)'^j form an integral base of the number field NF( n, stabilizer ), see Number Field Records.

super is a list representing a supergroup of the group described by the list stabilizer; the base is chosen such that the group of super acts on it, as far as this is possible.

Note: The b_i are in general not sets, since for stabilizer = super, b_i[1] is always an element of ZumbroichBase( N, 1 ); this is used by NF (see Number Field Records) and Coefficients (see Coefficients for Number Fields).

stabilizer must not contain the stabilizer of a proper cyclotomic subfield of Q_n.

    gap> LenstraBase( 24, [ 1, 19 ], [ 1, 19 ] );          # a base of
[ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]               # $Q_3(\sqrt{6})$,
gap> LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ] );   # another one
[ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
gap> LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ) ); # normal base of
[ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]          # $Q_3(\sqrt{5})$

GAP 3.4.4
April 1997