\\ Pari/GP code for working with number field 28.4.6026982035903235126420911997783559113692676096.3

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

\\ Define the number field: 
K = bnfinit(y^28 + 72*y^24 - 72*y^23 + 936*y^20 - 1920*y^19 - 288*y^18 + 1080*y^17 + 15930*y^16 + 5472*y^15 - 13680*y^14 - 6912*y^13 + 8208*y^12 + 97704*y^11 + 126336*y^10 + 77472*y^9 - 129384*y^8 - 228960*y^7 - 135792*y^6 - 83448*y^5 - 63639*y^4 - 43200*y^3 - 13104*y^2 - 1216*y + 576, 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^28 + 72*x^24 - 72*x^23 + 936*x^20 - 1920*x^19 - 288*x^18 + 1080*x^17 + 15930*x^16 + 5472*x^15 - 13680*x^14 - 6912*x^13 + 8208*x^12 + 97704*x^11 + 126336*x^10 + 77472*x^9 - 129384*x^8 - 228960*x^7 - 135792*x^6 - 83448*x^5 - 63639*x^4 - 43200*x^3 - 13104*x^2 - 1216*x + 576, 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]])