\\ Pari/GP code for working with number field 21.9.834507966212271820403374346664412679514757637054312831716289049278398513152.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^21 - 26208*y^19 - 195536*y^18 + 291973068*y^17 + 4409577342*y^16 - 1795665131924*y^15 - 41002370088498*y^14 + 6572542654341105*y^13 + 203858863109865284*y^12 - 14213279062376569485*y^11 - 582361703119054694040*y^10 + 16275661558142313035035*y^9 + 945745018043095393911180*y^8 - 5646302360560715152043103*y^7 - 792530253662409376623342416*y^6 - 4726551871709798292359565120*y^5 + 266740346394447654507510010344*y^4 + 2440298711745375561122234921808*y^3 - 44488899123058840085730493001568*y^2 - 315068960394756661920290849696640*y + 3724414958441731059478927509146752, 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^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752, 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 $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])