\\ Pari/GP code for working with number field 19.1.101575284882268140616515967431.1

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

\\ Define the number field: 
K = bnfinit(y^19 - 5*y^18 + 11*y^17 - 10*y^16 + 3*y^15 + 3*y^14 - 80*y^13 + 481*y^12 - 1180*y^11 + 1220*y^10 + 888*y^9 - 5070*y^8 + 8854*y^7 - 9440*y^6 + 6893*y^5 - 3748*y^4 + 2592*y^3 - 2790*y^2 + 2673*y - 1053, 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^19 - 5*x^18 + 11*x^17 - 10*x^16 + 3*x^15 + 3*x^14 - 80*x^13 + 481*x^12 - 1180*x^11 + 1220*x^10 + 888*x^9 - 5070*x^8 + 8854*x^7 - 9440*x^6 + 6893*x^5 - 3748*x^4 + 2592*x^3 - 2790*x^2 + 2673*x - 1053, 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]])