\\ Pari/GP code for working with number field 30.0.6072125144349637560639895035214067242224098130847.1

\\ 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 + 38*y^28 - 178*y^27 + 694*y^26 - 2325*y^25 + 6840*y^24 - 17890*y^23 + 42428*y^22 - 92148*y^21 + 185209*y^20 - 348333*y^19 + 617591*y^18 - 1037914*y^17 + 1659175*y^16 - 2519432*y^15 + 3620042*y^14 - 4892313*y^13 + 6168845*y^12 - 7195888*y^11 + 7698034*y^10 - 7475970*y^9 + 6524456*y^8 - 5050948*y^7 + 3416179*y^6 - 1976201*y^5 + 950326*y^4 - 363451*y^3 + 103267*y^2 - 19350*y + 1849, 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 + 38*x^28 - 178*x^27 + 694*x^26 - 2325*x^25 + 6840*x^24 - 17890*x^23 + 42428*x^22 - 92148*x^21 + 185209*x^20 - 348333*x^19 + 617591*x^18 - 1037914*x^17 + 1659175*x^16 - 2519432*x^15 + 3620042*x^14 - 4892313*x^13 + 6168845*x^12 - 7195888*x^11 + 7698034*x^10 - 7475970*x^9 + 6524456*x^8 - 5050948*x^7 + 3416179*x^6 - 1976201*x^5 + 950326*x^4 - 363451*x^3 + 103267*x^2 - 19350*x + 1849, 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]])