\\ Pari/GP code for working with number field 32.0.272292877590407567597186433188495333554574218609249.1

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

\\ Define the number field: 
K = bnfinit(y^32 - y^31 - y^30 + 3*y^29 - y^28 - 5*y^27 + 7*y^26 + 3*y^25 - 17*y^24 + 11*y^23 + 23*y^22 - 45*y^21 - y^20 + 91*y^19 - 89*y^18 - 93*y^17 + 271*y^16 - 186*y^15 - 356*y^14 + 728*y^13 - 16*y^12 - 1440*y^11 + 1472*y^10 + 1408*y^9 - 4352*y^8 + 1536*y^7 + 7168*y^6 - 10240*y^5 - 4096*y^4 + 24576*y^3 - 16384*y^2 - 32768*y + 65536, 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 - x^31 - x^30 + 3*x^29 - x^28 - 5*x^27 + 7*x^26 + 3*x^25 - 17*x^24 + 11*x^23 + 23*x^22 - 45*x^21 - x^20 + 91*x^19 - 89*x^18 - 93*x^17 + 271*x^16 - 186*x^15 - 356*x^14 + 728*x^13 - 16*x^12 - 1440*x^11 + 1472*x^10 + 1408*x^9 - 4352*x^8 + 1536*x^7 + 7168*x^6 - 10240*x^5 - 4096*x^4 + 24576*x^3 - 16384*x^2 - 32768*x + 65536, 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]])