\\ Pari/GP code for working with number field 33.33.27189028279553414235049966267283185807800188603627566700161.1

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

\\ Define the number field: 
K = bnfinit(y^33 - y^32 - 32*y^31 + 31*y^30 + 465*y^29 - 435*y^28 - 4060*y^27 + 3654*y^26 + 23751*y^25 - 20475*y^24 - 98280*y^23 + 80730*y^22 + 296010*y^21 - 230230*y^20 - 657800*y^19 + 480700*y^18 + 1081575*y^17 - 735471*y^16 - 1307504*y^15 + 817190*y^14 + 1144066*y^13 - 646646*y^12 - 705432*y^11 + 352716*y^10 + 293930*y^9 - 125970*y^8 - 77520*y^7 + 27132*y^6 + 11628*y^5 - 3060*y^4 - 816*y^3 + 136*y^2 + 17*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 - x^32 - 32*x^31 + 31*x^30 + 465*x^29 - 435*x^28 - 4060*x^27 + 3654*x^26 + 23751*x^25 - 20475*x^24 - 98280*x^23 + 80730*x^22 + 296010*x^21 - 230230*x^20 - 657800*x^19 + 480700*x^18 + 1081575*x^17 - 735471*x^16 - 1307504*x^15 + 817190*x^14 + 1144066*x^13 - 646646*x^12 - 705432*x^11 + 352716*x^10 + 293930*x^9 - 125970*x^8 - 77520*x^7 + 27132*x^6 + 11628*x^5 - 3060*x^4 - 816*x^3 + 136*x^2 + 17*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]])