\\ Pari/GP code for working with number field 27.1.3216045767164746225347277064349747511296.1

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

\\ Define the number field: 
K = bnfinit(y^27 - 10*y^26 + 38*y^25 - 52*y^24 - 84*y^23 + 458*y^22 - 715*y^21 + 40*y^20 + 1948*y^19 - 3756*y^18 + 1448*y^17 + 4952*y^16 - 7109*y^15 + 94*y^14 + 4978*y^13 - 2352*y^12 - 1300*y^11 + 2422*y^10 + 639*y^9 - 3848*y^8 - 28*y^7 + 1728*y^6 + 572*y^5 - 368*y^4 - 244*y^3 + 48*y^2 + 32*y - 16, 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 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16, 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]])