\\ Pari/GP code for working with number field 30.0.18201370075621419164235703647136688232421875.1

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

\\ Define the number field: 
K = bnfinit(y^30 - 10*y^27 - 54*y^25 + 90*y^24 + 315*y^23 + 15*y^22 - 240*y^21 + 504*y^20 - 4740*y^19 + 8375*y^18 - 14580*y^17 + 25140*y^16 - 34179*y^15 + 51465*y^14 - 61710*y^13 + 70215*y^12 - 67800*y^11 + 45276*y^10 - 22840*y^9 - 3240*y^8 + 16350*y^7 - 10730*y^6 + 4956*y^5 + 2280*y^4 - 4065*y^3 - 420*y^2 + 780*y + 169, 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 - 10*x^27 - 54*x^25 + 90*x^24 + 315*x^23 + 15*x^22 - 240*x^21 + 504*x^20 - 4740*x^19 + 8375*x^18 - 14580*x^17 + 25140*x^16 - 34179*x^15 + 51465*x^14 - 61710*x^13 + 70215*x^12 - 67800*x^11 + 45276*x^10 - 22840*x^9 - 3240*x^8 + 16350*x^7 - 10730*x^6 + 4956*x^5 + 2280*x^4 - 4065*x^3 - 420*x^2 + 780*x + 169, 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]])