\\ Pari/GP code for working with number field 33.33.2250468870721257864915422491944078011918280366020799086767808384626842687481.1

\\ 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 - 136*y^31 + 1010*y^30 + 9089*y^29 - 56260*y^28 - 391707*y^27 + 1778015*y^26 + 11776892*y^25 - 33684927*y^24 - 250499393*y^23 + 352249669*y^22 + 3723230595*y^21 - 869647969*y^20 - 37516024451*y^19 - 27213591295*y^18 + 241140895308*y^17 + 392273336476*y^16 - 846902299600*y^15 - 2477173011886*y^14 + 614677458641*y^13 + 7779541229922*y^12 + 5903210349213*y^11 - 9094078083973*y^10 - 16773542913789*y^9 - 4693562989906*y^8 + 8584245508513*y^7 + 7302900918860*y^6 + 488836631314*y^5 - 1567611092278*y^4 - 601455629303*y^3 - 17378174655*y^2 + 24259321620*y + 3109630591, 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 - 136*x^31 + 1010*x^30 + 9089*x^29 - 56260*x^28 - 391707*x^27 + 1778015*x^26 + 11776892*x^25 - 33684927*x^24 - 250499393*x^23 + 352249669*x^22 + 3723230595*x^21 - 869647969*x^20 - 37516024451*x^19 - 27213591295*x^18 + 241140895308*x^17 + 392273336476*x^16 - 846902299600*x^15 - 2477173011886*x^14 + 614677458641*x^13 + 7779541229922*x^12 + 5903210349213*x^11 - 9094078083973*x^10 - 16773542913789*x^9 - 4693562989906*x^8 + 8584245508513*x^7 + 7302900918860*x^6 + 488836631314*x^5 - 1567611092278*x^4 - 601455629303*x^3 - 17378174655*x^2 + 24259321620*x + 3109630591, 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]])