bnrgaloismatrix

Function: bnrgaloismatrix
Section: number_fields
C-Name: bnrgaloismatrix
Prototype: GG
Help: bnrgaloismatrix(bnr,aut): return the matrix of the action of the
 automorphism aut of the base field bnf.nf on the generators of the ray class
 field bnr.gen; aut can be given as a polynomial, or a vector of automorphisms
 or a galois group as output by galoisinit, in which case a vector of matrices
 is returned (in the later case, only for the generators aut.gen).
Doc: return the matrix of the action of the automorphism \var{aut} of the base
 field \kbd{bnf.nf} on the generators of the ray class field \kbd{bnr.gen};
 \var{aut} can be given as a polynomial, an algebraic number, or a vector of
 automorphisms or a Galois group as output by \kbd{galoisinit}, in which case a
 vector of matrices is returned (in the later case, only for the generators
 \kbd{aut.gen}).

 The generators \kbd{bnr.gen} need not be explicitly computed in the input
 \var{bnr}, which saves time: the result is well defined in this case also.

 \bprog
 ? K = bnfinit(a^4-3*a^2+253009); B = bnrinit(K,9); B.cyc
 %1 = [8400, 12, 6, 3]
 ? G = nfgaloisconj(K)
 %2 = [-a, a, -1/503*a^3 + 3/503*a, 1/503*a^3 - 3/503*a]~
 ? bnrgaloismatrix(B, G[2])  \\ G[2] = Id ...
 %3 =
 [1 0 0 0]

 [0 1 0 0]

 [0 0 1 0]

 [0 0 0 1]
 ? bnrgaloismatrix(B, G[3]) \\ automorphism of order 2
 %4 =
 [799 0 0 2800]

 [  0 7 0    4]

 [  4 0 5    2]

 [  0 0 0    2]
 ? M = %^2; for (i=1, #B.cyc, M[i,] %= B.cyc[i]); M
 %5 =  \\ acts on ray class group as automorphism of order 2
 [1 0 0 0]

 [0 1 0 0]

 [0 0 1 0]

 [0 0 0 1]
 @eprog

 See \kbd{bnrisgalois} for further examples.
Variant: When $aut$ is a polynomial or an algebraic number,
 \fun{GEN}{bnrautmatrix}{GEN bnr, GEN aut} is available.