\\ Pari/GP code for working with number field 36.0.17105807243512371509230217076280919792582365736923687236157265974689.1

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

\\ Define the number field: 
K = bnfinit(y^36 - 3*y^35 + 14*y^34 + 25*y^33 - 81*y^32 + 430*y^31 + 649*y^30 - 2959*y^29 + 10672*y^28 - 9183*y^27 + 3994*y^26 - 36672*y^25 + 129037*y^24 - 153463*y^23 - 377431*y^22 + 1507640*y^21 - 2832566*y^20 + 5098422*y^19 + 5245819*y^18 - 14119627*y^17 + 55722175*y^16 - 52547145*y^15 + 205342047*y^14 - 466686704*y^13 + 1314989350*y^12 - 1838473868*y^11 + 1895355654*y^10 - 622211603*y^9 - 531048524*y^8 + 1533170167*y^7 - 241708081*y^6 - 506053254*y^5 + 309479289*y^4 + 94941336*y^3 - 65292451*y^2 - 8212694*y + 10504081, 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^36 - 3*x^35 + 14*x^34 + 25*x^33 - 81*x^32 + 430*x^31 + 649*x^30 - 2959*x^29 + 10672*x^28 - 9183*x^27 + 3994*x^26 - 36672*x^25 + 129037*x^24 - 153463*x^23 - 377431*x^22 + 1507640*x^21 - 2832566*x^20 + 5098422*x^19 + 5245819*x^18 - 14119627*x^17 + 55722175*x^16 - 52547145*x^15 + 205342047*x^14 - 466686704*x^13 + 1314989350*x^12 - 1838473868*x^11 + 1895355654*x^10 - 622211603*x^9 - 531048524*x^8 + 1533170167*x^7 - 241708081*x^6 - 506053254*x^5 + 309479289*x^4 + 94941336*x^3 - 65292451*x^2 - 8212694*x + 10504081, 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]])