\\ Pari/GP code for working with number field 19.7.8322004363062341150231840635904.1

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

\\ Define the number field: 
K = bnfinit(y^19 - 3*y^18 - y^17 + 15*y^16 - 22*y^15 - 6*y^14 + 59*y^13 - 70*y^12 - 6*y^11 + 106*y^10 - 107*y^9 - y^8 + 92*y^7 - 72*y^6 - y^5 + 35*y^4 - 19*y^3 - y^2 + 4*y - 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

\\ Analytic class number formula: 
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - x^2 + 4*x - 1, 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]])