\\ Pari/GP code for working with number field 28.0.71403665296191917297019015087709113341743884283129.1

\\ 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 + 41*y^26 - 34*y^25 + 814*y^24 - 569*y^23 + 10073*y^22 - 5894*y^21 + 85446*y^20 - 41430*y^19 + 517766*y^18 - 204124*y^17 + 2278084*y^16 - 713493*y^15 + 7270037*y^14 - 1746022*y^13 + 16581844*y^12 - 2930752*y^11 + 26206720*y^10 - 3163168*y^9 + 27305280*y^8 - 2133504*y^7 + 17228288*y^6 - 594944*y^5 + 5748736*y^4 - 229376*y^3 + 745472*y^2 + 57344*y + 16384, 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 + 41*x^26 - 34*x^25 + 814*x^24 - 569*x^23 + 10073*x^22 - 5894*x^21 + 85446*x^20 - 41430*x^19 + 517766*x^18 - 204124*x^17 + 2278084*x^16 - 713493*x^15 + 7270037*x^14 - 1746022*x^13 + 16581844*x^12 - 2930752*x^11 + 26206720*x^10 - 3163168*x^9 + 27305280*x^8 - 2133504*x^7 + 17228288*x^6 - 594944*x^5 + 5748736*x^4 - 229376*x^3 + 745472*x^2 + 57344*x + 16384, 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]])