\\ Pari/GP code for working with number field 28.28.14603047886206093768209337615200705673567789821.1

\\ 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 - 28*y^26 + 28*y^25 + 349*y^24 - 349*y^23 - 2551*y^22 + 2551*y^21 + 12123*y^20 - 12123*y^19 - 39236*y^18 + 39236*y^17 + 88045*y^16 - 88045*y^15 - 136763*y^14 + 136763*y^13 + 144247*y^12 - 144247*y^11 - 99295*y^10 + 99295*y^9 + 41703*y^8 - 41703*y^7 - 9569*y^6 + 9569*y^5 + 987*y^4 - 987*y^3 - 28*y^2 + 28*y + 1, 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 - 28*x^26 + 28*x^25 + 349*x^24 - 349*x^23 - 2551*x^22 + 2551*x^21 + 12123*x^20 - 12123*x^19 - 39236*x^18 + 39236*x^17 + 88045*x^16 - 88045*x^15 - 136763*x^14 + 136763*x^13 + 144247*x^12 - 144247*x^11 - 99295*x^10 + 99295*x^9 + 41703*x^8 - 41703*x^7 - 9569*x^6 + 9569*x^5 + 987*x^4 - 987*x^3 - 28*x^2 + 28*x + 1, 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]])