\\ Pari/GP code for working with number field 25.1.10638544719000572788610677956885758729581201.1

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

\\ Define the number field: 
K = bnfinit(y^25 - 10*y^24 + 41*y^23 - 105*y^22 + 177*y^21 - 437*y^20 + 1470*y^19 - 3743*y^18 + 4598*y^17 - 1850*y^16 + 3734*y^15 - 33702*y^14 + 37801*y^13 + 70620*y^12 - 146863*y^11 - 115089*y^10 + 114649*y^9 + 185307*y^8 - 74232*y^7 - 239939*y^6 - 516736*y^5 - 486684*y^4 - 187312*y^3 - 162320*y^2 + 32064*y - 22784, 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 - 10*x^24 + 41*x^23 - 105*x^22 + 177*x^21 - 437*x^20 + 1470*x^19 - 3743*x^18 + 4598*x^17 - 1850*x^16 + 3734*x^15 - 33702*x^14 + 37801*x^13 + 70620*x^12 - 146863*x^11 - 115089*x^10 + 114649*x^9 + 185307*x^8 - 74232*x^7 - 239939*x^6 - 516736*x^5 - 486684*x^4 - 187312*x^3 - 162320*x^2 + 32064*x - 22784, 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]])