\\ Pari/GP code for working with number field 20.4.5117380860656444079832785566505699574289404854272.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 3*y^19 - 109*y^18 + 874*y^17 - 759*y^16 - 15157*y^15 - 135227*y^14 + 1527264*y^13 + 927943*y^12 - 29902077*y^11 + 15880149*y^10 + 910761676*y^9 - 3607945172*y^8 - 25051446848*y^7 + 47447295520*y^6 + 579288476848*y^5 + 469017755136*y^4 - 4937110551040*y^3 - 15529734627008*y^2 - 9459350206144*y + 6774058131520, 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^20 - 3*x^19 - 109*x^18 + 874*x^17 - 759*x^16 - 15157*x^15 - 135227*x^14 + 1527264*x^13 + 927943*x^12 - 29902077*x^11 + 15880149*x^10 + 910761676*x^9 - 3607945172*x^8 - 25051446848*x^7 + 47447295520*x^6 + 579288476848*x^5 + 469017755136*x^4 - 4937110551040*x^3 - 15529734627008*x^2 - 9459350206144*x + 6774058131520, 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]])