\\ Pari/GP code for working with number field 36.0.11349174172096312401159270887667863929976078528910955905024.1

\\ 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 - 9*y^34 + 45*y^33 - 6*y^32 - 423*y^31 + 987*y^30 + 1500*y^29 - 7605*y^28 + 1541*y^27 + 30849*y^26 - 54360*y^25 - 60048*y^24 + 185232*y^23 + 256266*y^22 - 152940*y^21 - 1073577*y^20 - 336708*y^19 + 2425303*y^18 + 1698282*y^17 - 2128815*y^16 - 3681504*y^15 - 275778*y^14 + 5895882*y^13 + 1320516*y^12 - 4471362*y^11 + 1316007*y^10 + 878499*y^9 - 806031*y^8 + 4374*y^7 + 235467*y^6 - 78003*y^5 - 16038*y^4 + 16767*y^3 - 2187*y^2 - 2187*y + 729, 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 - 9*x^34 + 45*x^33 - 6*x^32 - 423*x^31 + 987*x^30 + 1500*x^29 - 7605*x^28 + 1541*x^27 + 30849*x^26 - 54360*x^25 - 60048*x^24 + 185232*x^23 + 256266*x^22 - 152940*x^21 - 1073577*x^20 - 336708*x^19 + 2425303*x^18 + 1698282*x^17 - 2128815*x^16 - 3681504*x^15 - 275778*x^14 + 5895882*x^13 + 1320516*x^12 - 4471362*x^11 + 1316007*x^10 + 878499*x^9 - 806031*x^8 + 4374*x^7 + 235467*x^6 - 78003*x^5 - 16038*x^4 + 16767*x^3 - 2187*x^2 - 2187*x + 729, 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]])