\\ Pari/GP code for working with number field 25.25.338813178901720135627329000271856784820556640625.1

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

\\ Define the number field: 
K = bnfinit(y^25 - 50*y^23 + 1025*y^21 - 11250*y^19 - 125*y^18 + 72525*y^17 + 3100*y^16 - 283885*y^15 - 28375*y^14 + 674550*y^13 + 121800*y^12 - 942450*y^11 - 261005*y^10 + 718625*y^9 + 269475*y^8 - 258425*y^7 - 117125*y^6 + 33010*y^5 + 16625*y^4 - 1100*y^3 - 650*y^2 - 50*y - 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 - 50*x^23 + 1025*x^21 - 11250*x^19 - 125*x^18 + 72525*x^17 + 3100*x^16 - 283885*x^15 - 28375*x^14 + 674550*x^13 + 121800*x^12 - 942450*x^11 - 261005*x^10 + 718625*x^9 + 269475*x^8 - 258425*x^7 - 117125*x^6 + 33010*x^5 + 16625*x^4 - 1100*x^3 - 650*x^2 - 50*x - 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]])