\\ Pari/GP code for working with number field 32.0.6142666889587199870339155304469168186187744140625.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^32 - 4*y^31 + 38*y^30 - 130*y^29 + 699*y^28 - 2191*y^27 + 8745*y^26 - 25571*y^25 + 83386*y^24 - 222738*y^23 + 612551*y^22 - 1445800*y^21 + 3366820*y^20 - 6839883*y^19 + 13440516*y^18 - 23161945*y^17 + 38310126*y^16 - 55684805*y^15 + 77474162*y^14 - 94829103*y^13 + 110710129*y^12 - 113727761*y^11 + 110357640*y^10 - 94103904*y^9 + 74479117*y^8 - 51698016*y^7 + 32429638*y^6 - 17496051*y^5 + 7973462*y^4 - 3036857*y^3 + 868340*y^2 - 166271*y + 16531, 1)

\\ Defining polynomial: 
K.pol

\\ Degree over Q: 
poldegree(K.pol)

\\ Signature: 
K.sign

\\ Discriminant: 
K.disc

\\ Ramified primes: 
factor(abs(K.disc))[,1]~

\\ Integral basis: 
K.zk

\\ Class group: 
K.clgp

\\ Unit rank: 
K.fu

\\ Generator for roots of unity: 
K.tu[2]

\\ Fundamental units: 
K.fu

\\ Regulator: 
K.reg

\\ Analytic class number formula: 
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 - 4*x^31 + 38*x^30 - 130*x^29 + 699*x^28 - 2191*x^27 + 8745*x^26 - 25571*x^25 + 83386*x^24 - 222738*x^23 + 612551*x^22 - 1445800*x^21 + 3366820*x^20 - 6839883*x^19 + 13440516*x^18 - 23161945*x^17 + 38310126*x^16 - 55684805*x^15 + 77474162*x^14 - 94829103*x^13 + 110710129*x^12 - 113727761*x^11 + 110357640*x^10 - 94103904*x^9 + 74479117*x^8 - 51698016*x^7 + 32429638*x^6 - 17496051*x^5 + 7973462*x^4 - 3036857*x^3 + 868340*x^2 - 166271*x + 16531, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

\\ Intermediate fields: 
L = nfsubfields(K); L[2..length(b)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])