\\ Pari/GP code for working with number field 27.9.17717054310925604811052271173453197606912.2

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

\\ Define the number field: 
K = bnfinit(y^27 - 9*y^26 + 36*y^25 - 54*y^24 - 180*y^23 + 1296*y^22 - 3699*y^21 + 5184*y^20 + 1737*y^19 - 28278*y^18 + 81630*y^17 - 152073*y^16 + 206037*y^15 - 206640*y^14 + 143748*y^13 - 36954*y^12 - 70929*y^11 + 125325*y^10 - 106329*y^9 + 53154*y^8 - 10548*y^7 - 9594*y^6 + 13185*y^5 - 9180*y^4 + 4068*y^3 - 1053*y^2 - 207*y + 163, 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^27 - 9*x^26 + 36*x^25 - 54*x^24 - 180*x^23 + 1296*x^22 - 3699*x^21 + 5184*x^20 + 1737*x^19 - 28278*x^18 + 81630*x^17 - 152073*x^16 + 206037*x^15 - 206640*x^14 + 143748*x^13 - 36954*x^12 - 70929*x^11 + 125325*x^10 - 106329*x^9 + 53154*x^8 - 10548*x^7 - 9594*x^6 + 13185*x^5 - 9180*x^4 + 4068*x^3 - 1053*x^2 - 207*x + 163, 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]])