\\ Pari/GP code for working with number field 32.0.7669926418924454281216000000000000000000000000.1

\\ 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 + 10*y^30 - 20*y^29 + 34*y^28 - 32*y^27 - 8*y^26 + 112*y^25 - 308*y^24 + 608*y^23 - 880*y^22 + 944*y^21 - 536*y^20 - 672*y^19 + 2960*y^18 - 6176*y^17 + 9744*y^16 - 12352*y^15 + 11840*y^14 - 5376*y^13 - 8576*y^12 + 30208*y^11 - 56320*y^10 + 77824*y^9 - 78848*y^8 + 57344*y^7 - 8192*y^6 - 65536*y^5 + 139264*y^4 - 163840*y^3 + 163840*y^2 - 131072*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 - 4*x^31 + 10*x^30 - 20*x^29 + 34*x^28 - 32*x^27 - 8*x^26 + 112*x^25 - 308*x^24 + 608*x^23 - 880*x^22 + 944*x^21 - 536*x^20 - 672*x^19 + 2960*x^18 - 6176*x^17 + 9744*x^16 - 12352*x^15 + 11840*x^14 - 5376*x^13 - 8576*x^12 + 30208*x^11 - 56320*x^10 + 77824*x^9 - 78848*x^8 + 57344*x^7 - 8192*x^6 - 65536*x^5 + 139264*x^4 - 163840*x^3 + 163840*x^2 - 131072*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]])