\\ Pari/GP code for working with number field 27.1.86311832016540901180083101912199499488333854262271.1

\\ 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 + 51*y^25 - 157*y^24 + 381*y^23 + 94*y^22 - 4772*y^21 + 22385*y^20 - 19633*y^19 - 132957*y^18 + 462384*y^17 + 49228*y^16 - 4729835*y^15 + 16673567*y^14 - 15312382*y^13 - 102305328*y^12 + 361800096*y^11 - 196173035*y^10 - 890969729*y^9 + 1569260108*y^8 - 437613721*y^7 - 1278467664*y^6 + 2134193061*y^5 - 1922367953*y^4 + 1085805378*y^3 - 524572351*y^2 + 152402636*y - 24878621, 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 + 51*x^25 - 157*x^24 + 381*x^23 + 94*x^22 - 4772*x^21 + 22385*x^20 - 19633*x^19 - 132957*x^18 + 462384*x^17 + 49228*x^16 - 4729835*x^15 + 16673567*x^14 - 15312382*x^13 - 102305328*x^12 + 361800096*x^11 - 196173035*x^10 - 890969729*x^9 + 1569260108*x^8 - 437613721*x^7 - 1278467664*x^6 + 2134193061*x^5 - 1922367953*x^4 + 1085805378*x^3 - 524572351*x^2 + 152402636*x - 24878621, 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]])