\\ Pari/GP code for working with number field 25.5.9536743164062500000000000000000000.1

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

\\ Define the number field: 
K = bnfinit(y^25 - 5*y^24 + 10*y^23 - 10*y^22 + 17*y^20 - 25*y^19 + 10*y^18 + 55*y^17 - 150*y^16 + 156*y^15 - 100*y^14 + 115*y^13 - 60*y^12 - 175*y^11 + 208*y^10 + 70*y^9 - 80*y^8 - 70*y^7 - 25*y^6 + 20*y^5 + 15*y^4 + 15*y^3 + 10*y^2 - 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^25 - 5*x^24 + 10*x^23 - 10*x^22 + 17*x^20 - 25*x^19 + 10*x^18 + 55*x^17 - 150*x^16 + 156*x^15 - 100*x^14 + 115*x^13 - 60*x^12 - 175*x^11 + 208*x^10 + 70*x^9 - 80*x^8 - 70*x^7 - 25*x^6 + 20*x^5 + 15*x^4 + 15*x^3 + 10*x^2 - 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]])