\\ Pari/GP code for working with number field 18.0.3121575272477039793063598549762509621195780096.1

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

\\ Define the number field: 
K = bnfinit(y^18 - 3*y^17 + 33*y^16 - 388*y^15 + 3138*y^14 + 13218*y^13 + 53812*y^12 - 1464732*y^11 + 5349033*y^10 - 8306493*y^9 + 255408471*y^8 - 450926892*y^7 - 2241288864*y^6 - 4931594442*y^5 + 43187005062*y^4 + 351848838744*y^3 + 1835309384631*y^2 + 4396464030753*y + 7107138128187, 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 - 3*x^17 + 33*x^16 - 388*x^15 + 3138*x^14 + 13218*x^13 + 53812*x^12 - 1464732*x^11 + 5349033*x^10 - 8306493*x^9 + 255408471*x^8 - 450926892*x^7 - 2241288864*x^6 - 4931594442*x^5 + 43187005062*x^4 + 351848838744*x^3 + 1835309384631*x^2 + 4396464030753*x + 7107138128187, 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]])