\\ Pari/GP code for working with number field 28.0.1923732503543401200540313355166872100640336877.1

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

\\ Define the number field: 
K = bnfinit(y^28 - 12*y^27 + 74*y^26 - 288*y^25 + 773*y^24 - 1407*y^23 + 1601*y^22 - 676*y^21 - 794*y^20 + 2173*y^19 - 6644*y^18 + 23336*y^17 - 53512*y^16 + 82551*y^15 - 80880*y^14 + 23133*y^13 + 117766*y^12 - 327568*y^11 + 536090*y^10 - 646700*y^9 + 621936*y^8 - 488735*y^7 + 317878*y^6 - 166984*y^5 + 68952*y^4 - 21502*y^3 + 5711*y^2 - 1133*y + 193, 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 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193, 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]])