\\ Pari/GP code for working with number field 36.0.1078971204830304425585378172062472691337937003390705901956047673685421797.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^36 - 9*y^35 + 16*y^34 + 26*y^33 + 181*y^32 - 1445*y^31 + 2604*y^30 - 2755*y^29 + 9910*y^28 - 30401*y^27 + 82223*y^26 - 197798*y^25 + 452764*y^24 - 1456052*y^23 + 2709150*y^22 + 1162726*y^21 - 8566260*y^20 - 8520187*y^19 + 71726367*y^18 - 146575781*y^17 + 219113520*y^16 - 450921336*y^15 + 971562775*y^14 - 1564444576*y^13 + 2406821450*y^12 - 4029586993*y^11 + 5513868716*y^10 - 4668590893*y^9 + 1306426495*y^8 + 1351057679*y^7 - 782004588*y^6 - 648398298*y^5 + 17515236*y^4 + 769014408*y^3 + 45608456*y^2 - 600274541*y + 239345231, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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^36 - 9*x^35 + 16*x^34 + 26*x^33 + 181*x^32 - 1445*x^31 + 2604*x^30 - 2755*x^29 + 9910*x^28 - 30401*x^27 + 82223*x^26 - 197798*x^25 + 452764*x^24 - 1456052*x^23 + 2709150*x^22 + 1162726*x^21 - 8566260*x^20 - 8520187*x^19 + 71726367*x^18 - 146575781*x^17 + 219113520*x^16 - 450921336*x^15 + 971562775*x^14 - 1564444576*x^13 + 2406821450*x^12 - 4029586993*x^11 + 5513868716*x^10 - 4668590893*x^9 + 1306426495*x^8 + 1351057679*x^7 - 782004588*x^6 - 648398298*x^5 + 17515236*x^4 + 769014408*x^3 + 45608456*x^2 - 600274541*x + 239345231, 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]])