\\ Pari/GP code for working with number field 21.21.3919222256983996499704582149784632135677558495355107961.2.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^21 - 3*y^20 - 198*y^19 + 366*y^18 + 16005*y^17 - 9597*y^16 - 671202*y^15 - 438534*y^14 + 14998752*y^13 + 27946195*y^12 - 155865357*y^11 - 508309275*y^10 + 311371348*y^9 + 3072449295*y^8 + 3830424663*y^7 - 635242124*y^6 - 4505239431*y^5 - 2657813739*y^4 + 449118201*y^3 + 909084162*y^2 + 287915376*y + 27487873, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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^21 - 3*x^20 - 198*x^19 + 366*x^18 + 16005*x^17 - 9597*x^16 - 671202*x^15 - 438534*x^14 + 14998752*x^13 + 27946195*x^12 - 155865357*x^11 - 508309275*x^10 + 311371348*x^9 + 3072449295*x^8 + 3830424663*x^7 - 635242124*x^6 - 4505239431*x^5 - 2657813739*x^4 + 449118201*x^3 + 909084162*x^2 + 287915376*x + 27487873, 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(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])