\\ Pari/GP code for working with number field 28.2.2443365527501925442977500495872000000000000000000000.1

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

\\ Define the number field: 
K = bnfinit(y^28 + 10*y^26 - 280*y^24 - 2600*y^22 + 24800*y^20 + 288000*y^18 - 1128000*y^16 - 72400000*y^14 - 791360000*y^12 - 3704000000*y^10 - 17401600000*y^8 - 19584000000*y^6 - 96256000000*y^4 - 10240000000*y^2 - 90880000000, 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 + 10*x^26 - 280*x^24 - 2600*x^22 + 24800*x^20 + 288000*x^18 - 1128000*x^16 - 72400000*x^14 - 791360000*x^12 - 3704000000*x^10 - 17401600000*x^8 - 19584000000*x^6 - 96256000000*x^4 - 10240000000*x^2 - 90880000000, 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]])