\\ Pari/GP code for working with number field 30.0.10745322081507339108891258824483901499337171.1

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

\\ Define the number field: 
K = bnfinit(y^30 - y^29 - 4*y^28 + 2*y^27 + 14*y^26 - 18*y^25 - 22*y^24 + 80*y^23 + 2*y^22 - 292*y^21 + 107*y^20 + 966*y^19 - 524*y^18 - 1689*y^17 + 1837*y^16 + 1003*y^15 - 6747*y^14 + 4493*y^13 + 13850*y^12 - 15662*y^11 - 11779*y^10 + 28372*y^9 - 4752*y^8 - 31522*y^7 + 29972*y^6 + 5164*y^5 - 25670*y^4 + 15150*y^3 + 3000*y^2 - 8125*y + 3125, 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 - x^29 - 4*x^28 + 2*x^27 + 14*x^26 - 18*x^25 - 22*x^24 + 80*x^23 + 2*x^22 - 292*x^21 + 107*x^20 + 966*x^19 - 524*x^18 - 1689*x^17 + 1837*x^16 + 1003*x^15 - 6747*x^14 + 4493*x^13 + 13850*x^12 - 15662*x^11 - 11779*x^10 + 28372*x^9 - 4752*x^8 - 31522*x^7 + 29972*x^6 + 5164*x^5 - 25670*x^4 + 15150*x^3 + 3000*x^2 - 8125*x + 3125, 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]])