\\ Pari/GP code for working with number field 18.10.279992823820843547402730217.8.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^18 - 4*y^17 - 7*y^16 + 43*y^15 - 82*y^14 - 34*y^13 + 308*y^12 - 525*y^11 + 564*y^10 + 534*y^9 - 2063*y^8 + 2838*y^7 + 1297*y^6 - 4579*y^5 - 2106*y^4 + 223*y^3 + 1009*y^2 + 376*y - 41, 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 - 4*x^17 - 7*x^16 + 43*x^15 - 82*x^14 - 34*x^13 + 308*x^12 - 525*x^11 + 564*x^10 + 534*x^9 - 2063*x^8 + 2838*x^7 + 1297*x^6 - 4579*x^5 - 2106*x^4 + 223*x^3 + 1009*x^2 + 376*x - 41, 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]])