\\ Pari/GP code for working with number field 32.0.11721315531921426838644724774450299033325694662265349722529.2.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^32 - 14*y^31 + 90*y^30 - 354*y^29 + 1060*y^28 - 3122*y^27 + 9718*y^26 - 27548*y^25 + 68323*y^24 - 162420*y^23 + 388682*y^22 - 885302*y^21 + 1883356*y^20 - 3859590*y^19 + 7747823*y^18 - 14894504*y^17 + 27483236*y^16 - 48703184*y^15 + 83847191*y^14 - 137630031*y^13 + 219099037*y^12 - 329935717*y^11 + 485458653*y^10 - 661629418*y^9 + 894648443*y^8 - 1078358227*y^7 + 1337537651*y^6 - 1342929253*y^5 + 1540908056*y^4 - 1158250822*y^3 + 1251370063*y^2 - 517957627*y + 555187123, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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 - 14*x^31 + 90*x^30 - 354*x^29 + 1060*x^28 - 3122*x^27 + 9718*x^26 - 27548*x^25 + 68323*x^24 - 162420*x^23 + 388682*x^22 - 885302*x^21 + 1883356*x^20 - 3859590*x^19 + 7747823*x^18 - 14894504*x^17 + 27483236*x^16 - 48703184*x^15 + 83847191*x^14 - 137630031*x^13 + 219099037*x^12 - 329935717*x^11 + 485458653*x^10 - 661629418*x^9 + 894648443*x^8 - 1078358227*x^7 + 1337537651*x^6 - 1342929253*x^5 + 1540908056*x^4 - 1158250822*x^3 + 1251370063*x^2 - 517957627*x + 555187123, 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]])