\\ Pari/GP code for working with number field 32.0.2363147162130074593554078242531588550681346814117431640625.1.
\\ 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 - 18*y^30 + 72*y^29 + 311*y^28 - 1090*y^27 - 2862*y^26 + 8691*y^25 + 24872*y^24 - 66988*y^23 - 86403*y^22 + 207758*y^21 + 395325*y^20 - 595053*y^19 - 1119168*y^18 + 799235*y^17 + 3298870*y^16 - 4221332*y^15 + 4111698*y^14 - 7461943*y^13 + 5469898*y^12 + 2888264*y^11 - 9443335*y^10 + 9992412*y^9 - 3874794*y^8 - 4061403*y^7 + 6823432*y^6 - 2937855*y^5 - 1141251*y^4 + 2061220*y^3 - 477856*y^2 - 165184*y + 215296, 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 - 18*x^30 + 72*x^29 + 311*x^28 - 1090*x^27 - 2862*x^26 + 8691*x^25 + 24872*x^24 - 66988*x^23 - 86403*x^22 + 207758*x^21 + 395325*x^20 - 595053*x^19 - 1119168*x^18 + 799235*x^17 + 3298870*x^16 - 4221332*x^15 + 4111698*x^14 - 7461943*x^13 + 5469898*x^12 + 2888264*x^11 - 9443335*x^10 + 9992412*x^9 - 3874794*x^8 - 4061403*x^7 + 6823432*x^6 - 2937855*x^5 - 1141251*x^4 + 2061220*x^3 - 477856*x^2 - 165184*x + 215296, 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(L)]

\\ Galois group: 
polgalois(K.pol)

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