\\ Pari/GP code for working with number field 45.45.299215681303998835585125432825671967739342947202402933846152911778748517690473818220198154449462890625.1

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

\\ Define the number field: 
K = bnfinit(y^45 - 5*y^44 - 120*y^43 + 570*y^42 + 6560*y^41 - 29130*y^40 - 216795*y^39 + 884935*y^38 + 4846835*y^37 - 17873115*y^36 - 77748559*y^35 + 254358990*y^34 + 926664900*y^33 - 2637505030*y^32 - 8389061335*y^31 + 20318079721*y^30 + 58494148730*y^29 - 117428976980*y^28 - 316667682495*y^27 + 510052425895*y^26 + 1335050206763*y^25 - 1653394093375*y^24 - 4375338382740*y^23 + 3923851568060*y^22 + 11069977015865*y^21 - 6532787291841*y^20 - 21348397264600*y^19 + 6845293868685*y^18 + 30760204370920*y^17 - 2712180162090*y^16 - 32145143355065*y^15 - 3532224571660*y^14 + 23314933370325*y^13 + 6500465979650*y^12 - 10976198292280*y^11 - 4703287953351*y^10 + 3011509345140*y^9 + 1752830160915*y^8 - 400149323895*y^7 - 327698710405*y^6 + 19849150584*y^5 + 30510229725*y^4 + 7497705*y^3 - 1362119415*y^2 - 14453695*y + 22808701, 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^45 - 5*x^44 - 120*x^43 + 570*x^42 + 6560*x^41 - 29130*x^40 - 216795*x^39 + 884935*x^38 + 4846835*x^37 - 17873115*x^36 - 77748559*x^35 + 254358990*x^34 + 926664900*x^33 - 2637505030*x^32 - 8389061335*x^31 + 20318079721*x^30 + 58494148730*x^29 - 117428976980*x^28 - 316667682495*x^27 + 510052425895*x^26 + 1335050206763*x^25 - 1653394093375*x^24 - 4375338382740*x^23 + 3923851568060*x^22 + 11069977015865*x^21 - 6532787291841*x^20 - 21348397264600*x^19 + 6845293868685*x^18 + 30760204370920*x^17 - 2712180162090*x^16 - 32145143355065*x^15 - 3532224571660*x^14 + 23314933370325*x^13 + 6500465979650*x^12 - 10976198292280*x^11 - 4703287953351*x^10 + 3011509345140*x^9 + 1752830160915*x^8 - 400149323895*x^7 - 327698710405*x^6 + 19849150584*x^5 + 30510229725*x^4 + 7497705*x^3 - 1362119415*x^2 - 14453695*x + 22808701, 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]])