\\ Pari/GP code for working with number field 18.6.36644198070556426025390625.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^18 - 12*y^15 + 36*y^14 - 33*y^12 + 117*y^11 - 171*y^10 - 2*y^9 + 135*y^8 - 81*y^7 - 30*y^6 + 36*y^5 + 81*y^4 - 393*y^3 + 531*y^2 - 270*y + 44, 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^18 - 12*x^15 + 36*x^14 - 33*x^12 + 117*x^11 - 171*x^10 - 2*x^9 + 135*x^8 - 81*x^7 - 30*x^6 + 36*x^5 + 81*x^4 - 393*x^3 + 531*x^2 - 270*x + 44, 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 pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])