\\ Pari/GP code for working with number field 28.28.642581186433047492874742549394260203375244140625.1

\\ 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 - 40*y^26 + 35*y^25 + 667*y^24 - 497*y^23 - 6073*y^22 + 3733*y^21 + 33381*y^20 - 16356*y^19 - 116335*y^18 + 43955*y^17 + 264151*y^16 - 74631*y^15 - 396001*y^14 + 80966*y^13 + 391804*y^12 - 55684*y^11 - 251669*y^10 + 23519*y^9 + 101079*y^8 - 5634*y^7 - 23674*y^6 + 574*y^5 + 2842*y^4 + 28*y^3 - 133*y^2 - 7*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 - 40*x^26 + 35*x^25 + 667*x^24 - 497*x^23 - 6073*x^22 + 3733*x^21 + 33381*x^20 - 16356*x^19 - 116335*x^18 + 43955*x^17 + 264151*x^16 - 74631*x^15 - 396001*x^14 + 80966*x^13 + 391804*x^12 - 55684*x^11 - 251669*x^10 + 23519*x^9 + 101079*x^8 - 5634*x^7 - 23674*x^6 + 574*x^5 + 2842*x^4 + 28*x^3 - 133*x^2 - 7*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]])