\\ Pari/GP code for working with number field 28.0.537788304782666838980279974392697944146481853517.1

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

\\ Define the number field: 
K = bnfinit(y^28 - 24*y^26 + 286*y^24 - 2327*y^22 + 12825*y^20 - 40336*y^18 + 39135*y^16 + 152588*y^14 - 531606*y^12 + 381077*y^10 + 2209555*y^8 - 10321444*y^6 + 18143914*y^4 - 8922944*y^2 + 1423325, 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 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325, 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]])