\\ Pari/GP code for working with number field 27.1.1543319746516623033280478216838436483146079.1

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

\\ Define the number field: 
K = bnfinit(y^27 - 6*y^26 + 31*y^25 - 80*y^24 + 117*y^23 - 51*y^22 - 243*y^21 + 807*y^20 - 1350*y^19 + 966*y^18 + 1935*y^17 - 9345*y^16 + 23103*y^15 - 43548*y^14 + 68751*y^13 - 94575*y^12 + 115269*y^11 - 125268*y^10 + 121785*y^9 - 105567*y^8 + 81027*y^7 - 54357*y^6 + 31122*y^5 - 14664*y^4 + 5393*y^3 - 1350*y^2 + 179*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 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*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]])