\\ Pari/GP code for working with number field 28.0.18634854406558377293367533932433545897882080078125.1

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

\\ Define the number field: 
K = bnfinit(y^28 - y^27 + 30*y^26 - 30*y^25 + 407*y^24 - 407*y^23 + 3307*y^22 - 3307*y^21 + 17981*y^20 - 17981*y^19 + 69340*y^18 - 69340*y^17 + 196621*y^16 - 196621*y^15 + 421429*y^14 - 421429*y^13 + 702439*y^12 - 702439*y^11 + 945981*y^10 - 945981*y^9 + 1086979*y^8 - 1086979*y^7 + 1138251*y^6 - 1138251*y^5 + 1148807*y^4 - 1148807*y^3 + 1149822*y^2 - 1149822*y + 1149851, 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 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851, 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]])