\\ Pari/GP code for working with number field 27.1.3639553781467035763182087002112051895733239.1

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

\\ Define the number field: 
K = bnfinit(y^27 - 2*y^26 + y^25 + 18*y^24 + 107*y^23 + 180*y^22 + 332*y^21 + 290*y^20 + 295*y^19 - 179*y^18 - 772*y^17 - 1774*y^16 - 2092*y^15 - 2320*y^14 - 811*y^13 + 489*y^12 + 3551*y^11 + 5424*y^10 + 7617*y^9 + 6962*y^8 + 6380*y^7 + 2824*y^6 + 2291*y^5 - 521*y^4 + 623*y^3 - 221*y^2 + 207*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 - 2*x^26 + x^25 + 18*x^24 + 107*x^23 + 180*x^22 + 332*x^21 + 290*x^20 + 295*x^19 - 179*x^18 - 772*x^17 - 1774*x^16 - 2092*x^15 - 2320*x^14 - 811*x^13 + 489*x^12 + 3551*x^11 + 5424*x^10 + 7617*x^9 + 6962*x^8 + 6380*x^7 + 2824*x^6 + 2291*x^5 - 521*x^4 + 623*x^3 - 221*x^2 + 207*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]])