\\ Pari/GP code for working with number field 27.1.78637606867438430727852801672920631138199.1

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

\\ Define the number field: 
K = bnfinit(y^27 - y^26 + 6*y^25 - 31*y^24 + 97*y^23 - 269*y^22 + 599*y^21 - 994*y^20 + 1307*y^19 - 1298*y^18 + 592*y^17 + 817*y^16 - 2461*y^15 + 4001*y^14 - 4903*y^13 + 4315*y^12 - 2279*y^11 - 512*y^10 + 3872*y^9 - 6814*y^8 + 7826*y^7 - 7115*y^6 + 5689*y^5 - 3842*y^4 + 1951*y^3 - 627*y^2 + 101*y - 1, 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 - x^26 + 6*x^25 - 31*x^24 + 97*x^23 - 269*x^22 + 599*x^21 - 994*x^20 + 1307*x^19 - 1298*x^18 + 592*x^17 + 817*x^16 - 2461*x^15 + 4001*x^14 - 4903*x^13 + 4315*x^12 - 2279*x^11 - 512*x^10 + 3872*x^9 - 6814*x^8 + 7826*x^7 - 7115*x^6 + 5689*x^5 - 3842*x^4 + 1951*x^3 - 627*x^2 + 101*x - 1, 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]])