\\ Pari/GP code for working with number field 28.2.725917312055025700242151478457681268310546875.1

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

\\ Define the number field: 
K = bnfinit(y^28 - 5*y^27 - 3*y^26 + 57*y^25 - 13*y^24 - 420*y^23 + 259*y^22 + 2099*y^21 - 1147*y^20 - 7975*y^19 + 7844*y^18 + 15881*y^17 - 34395*y^16 - 17074*y^15 + 107194*y^14 - 103829*y^13 + 94772*y^12 - 84342*y^11 - 113950*y^10 + 177636*y^9 - 26899*y^8 + 2410*y^7 + 74025*y^6 - 120275*y^5 - 49775*y^4 + 101875*y^3 - 41875*y^2 + 31250*y - 15625, 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 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625, 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]])