\\ Pari/GP code for working with number field 30.30.254863675513583304379979153001217903140700917991797.1

\\ 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 - 30*y^28 + 30*y^27 + 404*y^26 - 404*y^25 - 3223*y^24 + 3223*y^23 + 16927*y^22 - 16927*y^21 - 61503*y^20 + 61503*y^19 + 158101*y^18 - 158101*y^17 - 288950*y^16 + 288950*y^15 + 371908*y^14 - 371908*y^13 - 329002*y^12 + 329002*y^11 + 191674*y^10 - 191674*y^9 - 68664*y^8 + 68664*y^7 + 13548*y^6 - 13548*y^5 - 1208*y^4 + 1208*y^3 + 32*y^2 - 32*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 - x^29 - 30*x^28 + 30*x^27 + 404*x^26 - 404*x^25 - 3223*x^24 + 3223*x^23 + 16927*x^22 - 16927*x^21 - 61503*x^20 + 61503*x^19 + 158101*x^18 - 158101*x^17 - 288950*x^16 + 288950*x^15 + 371908*x^14 - 371908*x^13 - 329002*x^12 + 329002*x^11 + 191674*x^10 - 191674*x^9 - 68664*x^8 + 68664*x^7 + 13548*x^6 - 13548*x^5 - 1208*x^4 + 1208*x^3 + 32*x^2 - 32*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]])