\\ Pari/GP code for working with number field 30.0.214224941509935982232692876412219596671526317019923.1

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

\\ Define the number field: 
K = bnfinit(y^30 - 6*y^29 + 7*y^28 + 64*y^27 - 269*y^26 + 169*y^25 + 1697*y^24 - 5426*y^23 + 2993*y^22 + 23602*y^21 - 68143*y^20 + 32587*y^19 + 255113*y^18 - 722972*y^17 + 638195*y^16 + 912175*y^15 - 3578530*y^14 + 6806701*y^13 - 13186196*y^12 + 20272003*y^11 - 4641245*y^10 - 65229024*y^9 + 170166391*y^8 - 258098865*y^7 + 369027093*y^6 - 495862722*y^5 + 471214449*y^4 - 297134325*y^3 + 130881204*y^2 - 35973639*y + 5861241, 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^30 - 6*x^29 + 7*x^28 + 64*x^27 - 269*x^26 + 169*x^25 + 1697*x^24 - 5426*x^23 + 2993*x^22 + 23602*x^21 - 68143*x^20 + 32587*x^19 + 255113*x^18 - 722972*x^17 + 638195*x^16 + 912175*x^15 - 3578530*x^14 + 6806701*x^13 - 13186196*x^12 + 20272003*x^11 - 4641245*x^10 - 65229024*x^9 + 170166391*x^8 - 258098865*x^7 + 369027093*x^6 - 495862722*x^5 + 471214449*x^4 - 297134325*x^3 + 130881204*x^2 - 35973639*x + 5861241, 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]])