\\ Pari/GP code for working with number field 17.17.2200187128095499475530336818113367454680001.1

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

\\ Define the number field: 
K = bnfinit(y^17 - y^16 - 208*y^15 - 17*y^14 + 15287*y^13 + 13881*y^12 - 487578*y^11 - 703261*y^10 + 6754359*y^9 + 10540902*y^8 - 41136753*y^7 - 57683825*y^6 + 92010954*y^5 + 95287840*y^4 - 17501435*y^3 - 25026156*y^2 - 563260*y + 1246103, 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^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103, 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]])