\\ Pari/GP code for working with number field 20.12.41041796771414445392922107616798059242325016576.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^20 - 264*y^18 - 976*y^17 + 26874*y^16 + 202704*y^15 - 909396*y^14 - 15075000*y^13 - 30990747*y^12 + 380528352*y^11 + 2672674146*y^10 + 3920991528*y^9 - 30769064512*y^8 - 203355815760*y^7 - 607022085426*y^6 - 1074952849288*y^5 - 1143465755040*y^4 - 636507402480*y^3 - 64156438152*y^2 + 99783624600*y + 31046369877, 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^20 - 264*x^18 - 976*x^17 + 26874*x^16 + 202704*x^15 - 909396*x^14 - 15075000*x^13 - 30990747*x^12 + 380528352*x^11 + 2672674146*x^10 + 3920991528*x^9 - 30769064512*x^8 - 203355815760*x^7 - 607022085426*x^6 - 1074952849288*x^5 - 1143465755040*x^4 - 636507402480*x^3 - 64156438152*x^2 + 99783624600*x + 31046369877, 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 pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])