\\ Pari/GP code for working with number field 33.33.228343593450302703244344174036290254973199912242460469577320120681.1

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

\\ Define the number field: 
K = bnfinit(y^33 - 8*y^32 - 48*y^31 + 488*y^30 + 889*y^29 - 13082*y^28 - 6681*y^27 + 203003*y^26 - 17780*y^25 - 2022483*y^24 + 832653*y^23 + 13572345*y^22 - 8326527*y^21 - 62657637*y^20 + 47704277*y^19 + 199663335*y^18 - 178099102*y^17 - 433344046*y^16 + 447058648*y^15 + 617907878*y^14 - 750473359*y^13 - 533823452*y^12 + 817138243*y^11 + 221279551*y^10 - 544192951*y^9 + 11597098*y^8 + 199867737*y^7 - 43176794*y^6 - 33036600*y^5 + 11713378*y^4 + 1505253*y^3 - 925381*y^2 + 72350*y + 1013, 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^33 - 8*x^32 - 48*x^31 + 488*x^30 + 889*x^29 - 13082*x^28 - 6681*x^27 + 203003*x^26 - 17780*x^25 - 2022483*x^24 + 832653*x^23 + 13572345*x^22 - 8326527*x^21 - 62657637*x^20 + 47704277*x^19 + 199663335*x^18 - 178099102*x^17 - 433344046*x^16 + 447058648*x^15 + 617907878*x^14 - 750473359*x^13 - 533823452*x^12 + 817138243*x^11 + 221279551*x^10 - 544192951*x^9 + 11597098*x^8 + 199867737*x^7 - 43176794*x^6 - 33036600*x^5 + 11713378*x^4 + 1505253*x^3 - 925381*x^2 + 72350*x + 1013, 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]])