\\ Pari/GP code for working with number field 30.0.141684422447224323645012862864040611366932987.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^30 - 5*y^29 + 14*y^28 - 37*y^27 + 83*y^26 - 152*y^25 + 241*y^24 - 335*y^23 + 465*y^22 - 620*y^21 + 812*y^20 - 1084*y^19 + 1348*y^18 - 1578*y^17 + 1736*y^16 - 1826*y^15 + 1986*y^14 - 2009*y^13 + 2111*y^12 - 2179*y^11 + 1810*y^10 - 1551*y^9 + 1376*y^8 - 959*y^7 + 887*y^6 - 765*y^5 + 462*y^4 - 285*y^3 + 146*y^2 - 13*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 - 5*x^29 + 14*x^28 - 37*x^27 + 83*x^26 - 152*x^25 + 241*x^24 - 335*x^23 + 465*x^22 - 620*x^21 + 812*x^20 - 1084*x^19 + 1348*x^18 - 1578*x^17 + 1736*x^16 - 1826*x^15 + 1986*x^14 - 2009*x^13 + 2111*x^12 - 2179*x^11 + 1810*x^10 - 1551*x^9 + 1376*x^8 - 959*x^7 + 887*x^6 - 765*x^5 + 462*x^4 - 285*x^3 + 146*x^2 - 13*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]])