\\ Pari/GP code for working with number field 27.1.3567989400462155147048339898163452848532229105663.1

\\ 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 + 80*y^25 - 447*y^24 + 1863*y^23 - 6240*y^22 + 15201*y^21 - 25896*y^20 + 26931*y^19 + 29922*y^18 - 2157*y^17 + 228*y^16 + 1150141*y^15 - 1553359*y^14 + 3361193*y^13 - 357518*y^12 - 4399792*y^11 - 5659177*y^10 - 7694789*y^9 - 6381628*y^8 + 25148690*y^7 + 37280366*y^6 + 10154644*y^5 - 4374482*y^4 + 5689374*y^3 + 3195477*y^2 - 2094606*y + 247941, 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 + 80*x^25 - 447*x^24 + 1863*x^23 - 6240*x^22 + 15201*x^21 - 25896*x^20 + 26931*x^19 + 29922*x^18 - 2157*x^17 + 228*x^16 + 1150141*x^15 - 1553359*x^14 + 3361193*x^13 - 357518*x^12 - 4399792*x^11 - 5659177*x^10 - 7694789*x^9 - 6381628*x^8 + 25148690*x^7 + 37280366*x^6 + 10154644*x^5 - 4374482*x^4 + 5689374*x^3 + 3195477*x^2 - 2094606*x + 247941, 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]])