60.3-2 LenstraBase
‣ 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 [ b 1 , b 2 , , b k ] subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 𝑘 [b_{1},b_{2},\ldots,b_{k}] of lists, each b i subscript 𝑏 𝑖 b_{i} consisting of integers such that the elements j b i subscript 𝑗 subscript 𝑏 𝑖 \sum_{{j\in b_{i}}} E(n) j 𝑗 {}^{j} 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 b i subscript 𝑏 𝑖 b_{i} 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 ? 3 ( 6 ) subscript ? 3 6 ?_{3}(\sqrt{{6}}) , the third result describes a normal basis of ? 3 ( 5 ) subscript ? 3 5 ?_{3}(\sqrt{{5}}) .