\\ Pari/GP code for working with number field 27.27.4737131966005480104201495460338050600482023844126105114809.8

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

\\ Define the number field: 
K = bnfinit(y^27 - 198*y^25 - 261*y^24 + 15957*y^23 + 39798*y^22 - 653067*y^21 - 2354373*y^20 + 13795839*y^19 + 68282040*y^18 - 128625003*y^17 - 1020022524*y^16 + 17217990*y^15 + 7566935193*y^14 + 7502170482*y^13 - 24774464097*y^12 - 43165902780*y^11 + 20935131912*y^10 + 75968172036*y^9 + 25647134895*y^8 - 29829689559*y^7 - 24029279100*y^6 - 5321891241*y^5 + 84272562*y^4 + 163197270*y^3 + 17369154*y^2 + 467856*y - 1083, 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^27 - 198*x^25 - 261*x^24 + 15957*x^23 + 39798*x^22 - 653067*x^21 - 2354373*x^20 + 13795839*x^19 + 68282040*x^18 - 128625003*x^17 - 1020022524*x^16 + 17217990*x^15 + 7566935193*x^14 + 7502170482*x^13 - 24774464097*x^12 - 43165902780*x^11 + 20935131912*x^10 + 75968172036*x^9 + 25647134895*x^8 - 29829689559*x^7 - 24029279100*x^6 - 5321891241*x^5 + 84272562*x^4 + 163197270*x^3 + 17369154*x^2 + 467856*x - 1083, 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]])