\\ Pari/GP code for working with number field 27.1.929766412363569678535445062868135961189021577764947.1

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

\\ Define the number field: 
K = bnfinit(y^27 - 11*y^26 + 7*y^25 + 393*y^24 - 1777*y^23 - 1339*y^22 + 33149*y^21 - 122161*y^20 + 301221*y^19 - 901165*y^18 + 2771309*y^17 - 5696089*y^16 + 7240134*y^15 - 11573472*y^14 + 45471960*y^13 - 153604016*y^12 + 357338624*y^11 - 615897728*y^10 + 876134656*y^9 - 1154442240*y^8 + 1414654976*y^7 - 1520924672*y^6 + 1526870016*y^5 - 1494863872*y^4 + 1216438272*y^3 - 684785664*y^2 + 248741888*y - 73596928, 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 - 11*x^26 + 7*x^25 + 393*x^24 - 1777*x^23 - 1339*x^22 + 33149*x^21 - 122161*x^20 + 301221*x^19 - 901165*x^18 + 2771309*x^17 - 5696089*x^16 + 7240134*x^15 - 11573472*x^14 + 45471960*x^13 - 153604016*x^12 + 357338624*x^11 - 615897728*x^10 + 876134656*x^9 - 1154442240*x^8 + 1414654976*x^7 - 1520924672*x^6 + 1526870016*x^5 - 1494863872*x^4 + 1216438272*x^3 - 684785664*x^2 + 248741888*x - 73596928, 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]])