\\ Pari/GP code for working with number field 27.27.2519933803975014472273213858760468555951745833259300216067364409.3.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^27 - 333*y^25 - 333*y^24 + 35298*y^23 + 53946*y^22 - 1819068*y^21 - 3588075*y^20 + 52704576*y^19 + 127734434*y^18 - 907552539*y^17 - 2673287370*y^16 + 9278172984*y^15 + 34236633762*y^14 - 51653439189*y^13 - 270237162441*y^12 + 93428282529*y^11 + 1280114382555*y^10 + 527095739526*y^9 - 3356233487919*y^8 - 3330019648665*y^7 + 3845500135071*y^6 + 6646340649357*y^5 + 111023372826*y^4 - 4196847578151*y^3 - 2127671869662*y^2 - 113904515466*y + 74779074553, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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 - 333*x^25 - 333*x^24 + 35298*x^23 + 53946*x^22 - 1819068*x^21 - 3588075*x^20 + 52704576*x^19 + 127734434*x^18 - 907552539*x^17 - 2673287370*x^16 + 9278172984*x^15 + 34236633762*x^14 - 51653439189*x^13 - 270237162441*x^12 + 93428282529*x^11 + 1280114382555*x^10 + 527095739526*x^9 - 3356233487919*x^8 - 3330019648665*x^7 + 3845500135071*x^6 + 6646340649357*x^5 + 111023372826*x^4 - 4196847578151*x^3 - 2127671869662*x^2 - 113904515466*x + 74779074553, 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(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])