\\ Pari/GP code for working with number field 27.3.995628422475629697764741523151987408896.1

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

\\ Define the number field: 
K = bnfinit(y^27 - 12*y^26 + 63*y^25 - 192*y^24 + 378*y^23 - 480*y^22 + 246*y^21 + 624*y^20 - 2346*y^19 + 4608*y^18 - 5598*y^17 + 2304*y^16 + 6564*y^15 - 19824*y^14 + 34248*y^13 - 40080*y^12 + 27606*y^11 - 2784*y^10 - 23718*y^9 + 45552*y^8 - 47970*y^7 + 29328*y^6 - 11814*y^5 + 4704*y^4 - 1779*y^3 + 468*y^2 - 105*y + 16, 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 - 12*x^26 + 63*x^25 - 192*x^24 + 378*x^23 - 480*x^22 + 246*x^21 + 624*x^20 - 2346*x^19 + 4608*x^18 - 5598*x^17 + 2304*x^16 + 6564*x^15 - 19824*x^14 + 34248*x^13 - 40080*x^12 + 27606*x^11 - 2784*x^10 - 23718*x^9 + 45552*x^8 - 47970*x^7 + 29328*x^6 - 11814*x^5 + 4704*x^4 - 1779*x^3 + 468*x^2 - 105*x + 16, 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]])