\\ Pari/GP code for working with number field 30.2.51586587308205615747892932945519983768463134765625.1

\\ 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^29 + 36*y^28 - 34*y^27 - 129*y^26 + 673*y^25 - 2345*y^24 + 3400*y^23 + 8195*y^22 - 23985*y^21 - 13829*y^20 + 36240*y^19 + 94321*y^18 + 102411*y^17 + 62946*y^16 + 325659*y^15 + 1299805*y^14 + 3826146*y^13 + 7335516*y^12 + 9580506*y^11 + 8499204*y^10 + 5247290*y^9 + 1937111*y^8 + 388946*y^7 - 47819*y^6 - 74039*y^5 - 43685*y^4 - 5057*y^3 + 3493*y^2 + 123*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^29 + 36*x^28 - 34*x^27 - 129*x^26 + 673*x^25 - 2345*x^24 + 3400*x^23 + 8195*x^22 - 23985*x^21 - 13829*x^20 + 36240*x^19 + 94321*x^18 + 102411*x^17 + 62946*x^16 + 325659*x^15 + 1299805*x^14 + 3826146*x^13 + 7335516*x^12 + 9580506*x^11 + 8499204*x^10 + 5247290*x^9 + 1937111*x^8 + 388946*x^7 - 47819*x^6 - 74039*x^5 - 43685*x^4 - 5057*x^3 + 3493*x^2 + 123*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]])