\\ Pari/GP code for working with number field 30.2.14026461290181639205847118647072000000000000000.1

\\ 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 + 11*y^28 + 4*y^27 + 44*y^26 - 6*y^25 - 74*y^24 - 446*y^23 - 1543*y^22 + 183*y^21 - 1721*y^20 + 7116*y^19 + 11884*y^18 - 3168*y^17 + 7664*y^16 - 18756*y^15 + 17109*y^14 - 3363*y^13 - 2431*y^12 + 11594*y^11 - 15080*y^10 + 6868*y^9 + 3884*y^8 - 3768*y^7 + 2512*y^6 - 1864*y^5 - 1696*y^4 + 656*y^3 + 448*y^2 + 16*y - 16, 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 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16, 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]])