\\ Pari/GP code for working with number field 18.6.51624177137754227336536682168566437749832086648167693412837925000000000000.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^18 - 6*y^17 - 79635*y^16 + 726016*y^15 - 21346492116*y^14 + 262385243232*y^13 - 222640923218036*y^12 + 5287125117175824*y^11 + 36115874586042869742*y^10 - 315574620417688738476*y^9 + 1064246929788629129834742*y^8 - 10094756489315953842318336*y^7 + 5555830520763339794029338844*y^6 - 66983200453112231194458550608*y^5 - 95464018377851901223533667416996*y^4 + 244705662148773213221767445905616*y^3 - 887010722785755909642099142950505815*y^2 + 1957760134295180518228084208239496514*y - 371023633673372769266300083152445623019, 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 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019, 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]])