\\ Pari/GP code for working with number field 25.1.1219471702607488515548387495680371337890625.1

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

\\ Define the number field: 
K = bnfinit(y^25 - 11*y^24 + 42*y^23 - 37*y^22 - 156*y^21 + 345*y^20 + 267*y^19 - 2248*y^18 + 5531*y^17 - 338*y^16 - 41434*y^15 + 71882*y^14 + 112392*y^13 - 407865*y^12 + 80824*y^11 + 717928*y^10 - 228258*y^9 - 1185331*y^8 + 280801*y^7 + 1733675*y^6 - 74160*y^5 - 2202750*y^4 - 202350*y^3 + 1981500*y^2 + 536000*y - 1401875, 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 - 11*x^24 + 42*x^23 - 37*x^22 - 156*x^21 + 345*x^20 + 267*x^19 - 2248*x^18 + 5531*x^17 - 338*x^16 - 41434*x^15 + 71882*x^14 + 112392*x^13 - 407865*x^12 + 80824*x^11 + 717928*x^10 - 228258*x^9 - 1185331*x^8 + 280801*x^7 + 1733675*x^6 - 74160*x^5 - 2202750*x^4 - 202350*x^3 + 1981500*x^2 + 536000*x - 1401875, 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]])