\\ Pari/GP code for working with number field 27.1.119615770666944050013402329147269064161944026933311.1

\\ 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 + 56*y^25 - 273*y^24 + 1290*y^23 - 5604*y^22 + 24003*y^21 - 98346*y^20 + 381168*y^19 - 1383408*y^18 + 4700958*y^17 - 15265944*y^16 + 47759673*y^15 - 142609974*y^14 + 403676163*y^13 - 1071234237*y^12 + 2616260829*y^11 - 5766916275*y^10 + 11330239941*y^9 - 19648861461*y^8 + 29452821249*y^7 - 36679431165*y^6 + 35809916697*y^5 - 25389897726*y^4 + 11811942110*y^3 - 3328128234*y^2 + 1063555192*y - 565863243, 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 + 56*x^25 - 273*x^24 + 1290*x^23 - 5604*x^22 + 24003*x^21 - 98346*x^20 + 381168*x^19 - 1383408*x^18 + 4700958*x^17 - 15265944*x^16 + 47759673*x^15 - 142609974*x^14 + 403676163*x^13 - 1071234237*x^12 + 2616260829*x^11 - 5766916275*x^10 + 11330239941*x^9 - 19648861461*x^8 + 29452821249*x^7 - 36679431165*x^6 + 35809916697*x^5 - 25389897726*x^4 + 11811942110*x^3 - 3328128234*x^2 + 1063555192*x - 565863243, 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]])