\\ Pari/GP code for working with number field 30.0.8851375005191383462430321349782942769050796640392327.1

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

\\ Define the number field: 
K = bnfinit(y^30 - 4*y^29 - 32*y^28 + 92*y^27 + 588*y^26 - 752*y^25 - 6717*y^24 - 805*y^23 + 44049*y^22 + 54263*y^21 - 93757*y^20 - 350454*y^19 - 591742*y^18 + 307201*y^17 + 3600412*y^16 + 5669890*y^15 - 632841*y^14 - 17605194*y^13 - 34361042*y^12 - 23087625*y^11 + 45815379*y^10 + 119941341*y^9 + 85868062*y^8 - 30516460*y^7 - 62011703*y^6 + 46150369*y^5 + 155790202*y^4 + 147686157*y^3 + 67768188*y^2 + 10788948*y + 751689, 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 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689, 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]])