\\ Pari/GP code for working with number field 18.18.389573518709461277048874506777900581813141954677569649354154692963827395433931232826260909975966492327717495503836697626432438146795529306112.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^18 - 146687112*y^16 - 37892719296*y^15 + 8408618911888512*y^14 + 4325813678364681744*y^13 - 243604861180754026285992*y^12 - 192688273988213296086844512*y^11 + 3797668582351192771935287035158*y^10 + 4155203541072329873982601967541328*y^9 - 30824703601196378502940767667441818360*y^8 - 44117476415146499319729949379490251620992*y^7 + 112503359565037178406650855732715578314837728*y^6 + 209476226244754241024858216530145737787244951984*y^5 - 121787795931109896131166351794082151507555501395576*y^4 - 371916116695980755216640900185153166330681546329851680*y^3 - 57935826570177631099916455044945775522459483676836854543*y^2 + 201710684116538249174185560612387919784188057130327410730544*y + 93572448667401892834025834694441537328078001525771364657373312, 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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312, 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]])