\\ Pari/GP code for working with number field 25.1.1149142693715820345877000392469173818401.1

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

\\ Define the number field: 
K = bnfinit(y^25 - 4*y^24 + y^23 + 2*y^22 + 5*y^21 + 38*y^20 + 24*y^19 - 79*y^18 - 45*y^17 - 175*y^16 - 246*y^15 + 78*y^14 + 160*y^13 + 28*y^12 + 1032*y^11 + 953*y^10 + 967*y^9 + 1690*y^8 + 1081*y^7 - 1035*y^6 - 715*y^5 - 2390*y^4 - 2966*y^3 - 1925*y^2 - 720*y - 1003, 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 - 4*x^24 + x^23 + 2*x^22 + 5*x^21 + 38*x^20 + 24*x^19 - 79*x^18 - 45*x^17 - 175*x^16 - 246*x^15 + 78*x^14 + 160*x^13 + 28*x^12 + 1032*x^11 + 953*x^10 + 967*x^9 + 1690*x^8 + 1081*x^7 - 1035*x^6 - 715*x^5 - 2390*x^4 - 2966*x^3 - 1925*x^2 - 720*x - 1003, 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]])