\\ Pari/GP code for working with number field 19.1.49578494467761916312526550343.1

\\ (Note that not all these functions may be available, and some may take a long time to execute.)

\\ Define the number field: 
K = bnfinit(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*x - 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

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
p = 7;  \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
idealfactors = idealprimedec(K, p); \\ get the data
vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])