Check this out sm88,
(17:57) gp > polsubcyclo(2201, 5)
%33 = [x^5  x^4  880*x^3 + 176*x^2 + 179584*x + 26624, x^5 + x^4  28*x^3 + 37*x^2 + 25*x + 1, x^5  x^4  880*x^3 + 6779*x^2 + 14509*x  112039, x^5 + x^4  12*x^3  21*x^2 + x + 5, x^5  x^4  880*x^3 + 15583*x^2  95541*x + 196101, x^5  x^4  880*x^3  2025*x^2 + 49725*x  112039]
I figured out the idea to this: polsubcyclo(n, d) will give ALL polynomials with the same number field as the cyclotomic polynomial d if and only if d  phi(n). I've organized the new polynomials. Dr. Sardonicus should take a look at these ones:
x^5  x^4  880*x^3 + 176*x^2 + 179584*x + 26624
x^5 + x^4  28*x^3 + 37*x^2 + 25*x + 1
x^5  x^4  880*x^3 + 6779*x^2 + 14509*x  112039,
x^5 + x^4  12*x^3  21*x^2 + x + 5
x^5  x^4  880*x^3 + 15583*x^2  95541*x + 196101
x^5  x^4  880*x^3  2025*x^2 + 49725*x  112039
All 5 appear to have the same number field properties as x^5+x^4+x^3+x^2+x+1
EDIT:
The mistake is the polynomials are degree 5, not 4!
Command:
(17:54) gp > ?? polsubcyclo()
polsubcyclo(n,d,{v = 'x}):
Gives polynomials (in variable v) defining the subAbelian extensions of
degree d of the cyclotomic field Q(zeta_n), where d  phi(n).
If there is exactly one such extension the output is a polynomial, else it
is a vector of polynomials, possibly empty. To get a vector in all cases, use
concat([], polsubcyclo(n,d)).
The function galoissubcyclo allows to specify exactly which subAbelian
extension should be computed.
The library syntax is GEN polsubcyclo(long n, long d, long v = 1), where v
is a variable number.
Last fiddled with by carpetpool on 20170217 at 02:15
