\\ Pari/GP code for working with number field 28.2.2657211910834657108824251865653514862060546875.1

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

\\ Define the number field: 
K = bnfinit(y^28 - 9*y^27 + 53*y^26 - 226*y^25 + 695*y^24 - 1455*y^23 + 1352*y^22 + 4587*y^21 - 26437*y^20 + 75511*y^19 - 148243*y^18 + 166438*y^17 - 8850*y^16 - 553583*y^15 + 1599191*y^14 - 2514414*y^13 + 2904360*y^12 - 819447*y^11 - 1193138*y^10 + 5517125*y^9 - 17624126*y^8 + 7384001*y^7 - 17552991*y^6 + 11860646*y^5 - 8713168*y^4 - 7406098*y^3 - 9688274*y^2 - 1853761*y - 465059, 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 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059, 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]])