\\ Pari/GP code for working with number field 33.33.70011645999218458416472683122408534303895571350166174758601569.1

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

\\ Define the number field: 
K = bnfinit(y^33 - 63*y^31 - 4*y^30 + 1710*y^29 + 204*y^28 - 26254*y^27 - 4392*y^26 + 251754*y^25 + 52248*y^24 - 1572192*y^23 - 377544*y^22 + 6479940*y^21 + 1717920*y^20 - 17548167*y^19 - 4953616*y^18 + 30725595*y^17 + 8949468*y^16 - 34072421*y^15 - 9933084*y^14 + 23552382*y^13 + 6584132*y^12 - 10086444*y^11 - 2580438*y^10 + 2636960*y^9 + 587106*y^8 - 404862*y^7 - 74061*y^6 + 33768*y^5 + 4755*y^4 - 1326*y^3 - 135*y^2 + 18*y + 1, 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 - 63*x^31 - 4*x^30 + 1710*x^29 + 204*x^28 - 26254*x^27 - 4392*x^26 + 251754*x^25 + 52248*x^24 - 1572192*x^23 - 377544*x^22 + 6479940*x^21 + 1717920*x^20 - 17548167*x^19 - 4953616*x^18 + 30725595*x^17 + 8949468*x^16 - 34072421*x^15 - 9933084*x^14 + 23552382*x^13 + 6584132*x^12 - 10086444*x^11 - 2580438*x^10 + 2636960*x^9 + 587106*x^8 - 404862*x^7 - 74061*x^6 + 33768*x^5 + 4755*x^4 - 1326*x^3 - 135*x^2 + 18*x + 1, 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]])