\\ Pari/GP code for working with number field 30.0.45711723244122563996456104229882472771.1

\\ 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^28 - y^27 + 14*y^26 - y^25 + 8*y^24 + 13*y^23 - 39*y^22 + 37*y^21 - 93*y^20 + 30*y^19 + 10*y^18 - 13*y^17 + 349*y^16 + 77*y^15 + 721*y^14 + 575*y^13 + 1026*y^12 + 1233*y^11 + 1137*y^10 + 993*y^9 + 501*y^8 - 41*y^7 - 253*y^6 - 186*y^5 - 58*y^4 + 6*y^3 + 14*y^2 + 6*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^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*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]])