\\ Pari/GP code for working with number field 27.27.1670605670104664083337543234069150946215895123101528358912.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^27 - 78*y^25 - 69*y^24 + 2391*y^23 + 3729*y^22 - 36776*y^21 - 78168*y^20 + 303360*y^19 + 833056*y^18 - 1289328*y^17 - 4920498*y^16 + 1998961*y^15 + 16364808*y^14 + 3861165*y^13 - 29580077*y^12 - 18957114*y^11 + 26856084*y^10 + 25541057*y^9 - 11892138*y^8 - 16024680*y^7 + 2411455*y^6 + 5218554*y^5 - 217893*y^4 - 866850*y^3 + 37389*y^2 + 59427*y - 7561, 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 - 78*x^25 - 69*x^24 + 2391*x^23 + 3729*x^22 - 36776*x^21 - 78168*x^20 + 303360*x^19 + 833056*x^18 - 1289328*x^17 - 4920498*x^16 + 1998961*x^15 + 16364808*x^14 + 3861165*x^13 - 29580077*x^12 - 18957114*x^11 + 26856084*x^10 + 25541057*x^9 - 11892138*x^8 - 16024680*x^7 + 2411455*x^6 + 5218554*x^5 - 217893*x^4 - 866850*x^3 + 37389*x^2 + 59427*x - 7561, 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]])