\\ Pari/GP code for working with number field 30.0.38068577552393333958638260901553477592868939543059.1

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

\\ Define the number field: 
K = bnfinit(y^30 - 6*y^29 + 25*y^28 - 72*y^27 + 234*y^26 - 520*y^25 + 1111*y^24 - 2189*y^23 + 3903*y^22 - 4957*y^21 + 4623*y^20 - 10333*y^19 + 8702*y^18 + 8683*y^17 - 6137*y^16 + 9595*y^15 - 28557*y^14 + 23199*y^13 - 20007*y^12 + 4195*y^11 + 12906*y^10 - 27137*y^9 + 36711*y^8 - 24629*y^7 + 24367*y^6 - 23558*y^5 + 11264*y^4 + 26568*y^3 - 14144*y^2 - 4128*y + 4928, 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 - 6*x^29 + 25*x^28 - 72*x^27 + 234*x^26 - 520*x^25 + 1111*x^24 - 2189*x^23 + 3903*x^22 - 4957*x^21 + 4623*x^20 - 10333*x^19 + 8702*x^18 + 8683*x^17 - 6137*x^16 + 9595*x^15 - 28557*x^14 + 23199*x^13 - 20007*x^12 + 4195*x^11 + 12906*x^10 - 27137*x^9 + 36711*x^8 - 24629*x^7 + 24367*x^6 - 23558*x^5 + 11264*x^4 + 26568*x^3 - 14144*x^2 - 4128*x + 4928, 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]])