\\ Pari/GP code for working with number field 30.0.893083200934004322334040290940494739136293593671.1

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

\\ Define the number field: 
K = bnfinit(y^30 - 3*y^29 - 5*y^28 + 23*y^27 - 7*y^26 - 43*y^25 + 380*y^24 - 1004*y^23 - 505*y^22 + 4147*y^21 - 3960*y^20 + 145*y^19 + 33797*y^18 - 104422*y^17 + 76694*y^16 + 164546*y^15 - 470105*y^14 + 125016*y^13 + 1685293*y^12 - 1720079*y^11 - 1513677*y^10 - 5549753*y^9 + 23270864*y^8 - 21911244*y^7 + 5984424*y^6 - 15339818*y^5 + 45492605*y^4 - 53109297*y^3 + 35921397*y^2 - 13035410*y + 2348809, 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 - 3*x^29 - 5*x^28 + 23*x^27 - 7*x^26 - 43*x^25 + 380*x^24 - 1004*x^23 - 505*x^22 + 4147*x^21 - 3960*x^20 + 145*x^19 + 33797*x^18 - 104422*x^17 + 76694*x^16 + 164546*x^15 - 470105*x^14 + 125016*x^13 + 1685293*x^12 - 1720079*x^11 - 1513677*x^10 - 5549753*x^9 + 23270864*x^8 - 21911244*x^7 + 5984424*x^6 - 15339818*x^5 + 45492605*x^4 - 53109297*x^3 + 35921397*x^2 - 13035410*x + 2348809, 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]])