\\ Pari/GP code for working with number field 20.0.75045768722673667967319488525390625.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 5*y^19 + 5*y^18 + 30*y^17 - 460*y^16 + 1594*y^15 - 660*y^14 - 6250*y^13 + 56490*y^12 - 105110*y^11 + 59367*y^10 + 111430*y^9 - 1315060*y^8 + 1129650*y^7 + 448205*y^6 + 7848131*y^5 + 15953485*y^4 - 43775185*y^3 - 32489355*y^2 + 70381245*y + 55238851, 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 - 5*x^19 + 5*x^18 + 30*x^17 - 460*x^16 + 1594*x^15 - 660*x^14 - 6250*x^13 + 56490*x^12 - 105110*x^11 + 59367*x^10 + 111430*x^9 - 1315060*x^8 + 1129650*x^7 + 448205*x^6 + 7848131*x^5 + 15953485*x^4 - 43775185*x^3 - 32489355*x^2 + 70381245*x + 55238851, 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]])