\\ Pari/GP code for working with number field 25.1.39753813747116100568718553941184504222638641.1

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

\\ Define the number field: 
K = bnfinit(y^25 - y^24 + 2*y^23 - 2*y^22 + 114*y^21 + 14*y^20 + 388*y^19 + 1085*y^18 + 7513*y^17 + 4819*y^16 + 1219*y^15 - 640*y^14 + 110085*y^13 + 396579*y^12 + 932255*y^11 + 3052375*y^10 + 5536096*y^9 + 12433767*y^8 + 17351055*y^7 + 27726417*y^6 + 30856797*y^5 + 36500635*y^4 + 30496620*y^3 + 26587890*y^2 + 14289723*y + 8362683, 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 - x^24 + 2*x^23 - 2*x^22 + 114*x^21 + 14*x^20 + 388*x^19 + 1085*x^18 + 7513*x^17 + 4819*x^16 + 1219*x^15 - 640*x^14 + 110085*x^13 + 396579*x^12 + 932255*x^11 + 3052375*x^10 + 5536096*x^9 + 12433767*x^8 + 17351055*x^7 + 27726417*x^6 + 30856797*x^5 + 36500635*x^4 + 30496620*x^3 + 26587890*x^2 + 14289723*x + 8362683, 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]])