\\ Pari/GP code for working with number field 30.0.11277272245002111679540357002131401262707502951787.1

\\ 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 + 23*y^28 - 12*y^27 + 335*y^26 - 144*y^25 + 2773*y^24 - 863*y^23 + 16295*y^22 - 4775*y^21 + 62257*y^20 - 15750*y^19 + 170334*y^18 - 52802*y^17 + 293956*y^16 - 111720*y^15 + 369164*y^14 - 135917*y^13 + 276687*y^12 - 78965*y^11 + 139349*y^10 - 31906*y^9 + 42847*y^8 - 4541*y^7 + 8009*y^6 - 477*y^5 + 1036*y^4 + 2*y^3 + 63*y^2 - 6*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 + 23*x^28 - 12*x^27 + 335*x^26 - 144*x^25 + 2773*x^24 - 863*x^23 + 16295*x^22 - 4775*x^21 + 62257*x^20 - 15750*x^19 + 170334*x^18 - 52802*x^17 + 293956*x^16 - 111720*x^15 + 369164*x^14 - 135917*x^13 + 276687*x^12 - 78965*x^11 + 139349*x^10 - 31906*x^9 + 42847*x^8 - 4541*x^7 + 8009*x^6 - 477*x^5 + 1036*x^4 + 2*x^3 + 63*x^2 - 6*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]])