\\ Pari/GP code for working with number field 21.3.6055813052022442741267437386720431302520208823407199686319727.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^21 - 7*y^20 + 21*y^19 - 34*y^18 + 5335*y^17 - 68226*y^16 + 370487*y^15 - 976238*y^14 + 9992983*y^13 - 167147434*y^12 + 1254295567*y^11 - 5119024426*y^10 + 16406088995*y^9 - 124794841242*y^8 + 1235524808890*y^7 - 7765632654107*y^6 + 28259771083054*y^5 - 48713312542188*y^4 - 12644866212328*y^3 + 193183257790096*y^2 - 276127419440096*y + 122588921675584, 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 - 7*x^20 + 21*x^19 - 34*x^18 + 5335*x^17 - 68226*x^16 + 370487*x^15 - 976238*x^14 + 9992983*x^13 - 167147434*x^12 + 1254295567*x^11 - 5119024426*x^10 + 16406088995*x^9 - 124794841242*x^8 + 1235524808890*x^7 - 7765632654107*x^6 + 28259771083054*x^5 - 48713312542188*x^4 - 12644866212328*x^3 + 193183257790096*x^2 - 276127419440096*x + 122588921675584, 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]])