\\ Pari/GP code for working with number field 32.0.2294081869728198830572049522437155246734619140625.1

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

\\ Define the number field: 
K = bnfinit(y^32 - 11*y^31 + 68*y^30 - 336*y^29 + 1315*y^28 - 4335*y^27 + 12498*y^26 - 30713*y^25 + 67152*y^24 - 127199*y^23 + 211471*y^22 - 307078*y^21 + 387026*y^20 - 419617*y^19 + 359728*y^18 - 149755*y^17 - 229142*y^16 + 581095*y^15 - 441722*y^14 - 315448*y^13 + 1021466*y^12 - 946897*y^11 + 366311*y^10 - 50246*y^9 + 132237*y^8 - 215882*y^7 + 157773*y^6 - 97500*y^5 + 79930*y^4 - 54894*y^3 + 25063*y^2 - 7709*y + 3131, 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 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131, 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]])