\\ Pari/GP code for working with number field 44.44.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.1

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

\\ Define the number field: 
K = bnfinit(y^44 - y^43 - 67*y^42 + 66*y^41 + 2006*y^40 - 1940*y^39 - 35522*y^38 + 33582*y^37 + 415391*y^36 - 381809*y^35 - 3396482*y^34 + 3014673*y^33 + 20088042*y^32 - 17073369*y^31 - 87848856*y^30 + 70775487*y^29 + 288431096*y^28 - 217655609*y^27 - 718686410*y^26 + 501030801*y^25 + 1368730205*y^24 - 867699404*y^23 - 1999332041*y^22 + 1131696716*y^21 + 2238799988*y^20 - 1107666220*y^19 - 1911947202*y^18 + 806431252*y^17 + 1231950206*y^16 - 430200535*y^15 - 588270804*y^14 + 164470157*y^13 + 202516659*y^12 - 43750410*y^11 - 48240456*y^10 + 7822102*y^9 + 7480922*y^8 - 908640*y^7 - 688078*y^6 + 65581*y^5 + 31906*y^4 - 2504*y^3 - 504*y^2 + 24*y + 1, 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^44 - x^43 - 67*x^42 + 66*x^41 + 2006*x^40 - 1940*x^39 - 35522*x^38 + 33582*x^37 + 415391*x^36 - 381809*x^35 - 3396482*x^34 + 3014673*x^33 + 20088042*x^32 - 17073369*x^31 - 87848856*x^30 + 70775487*x^29 + 288431096*x^28 - 217655609*x^27 - 718686410*x^26 + 501030801*x^25 + 1368730205*x^24 - 867699404*x^23 - 1999332041*x^22 + 1131696716*x^21 + 2238799988*x^20 - 1107666220*x^19 - 1911947202*x^18 + 806431252*x^17 + 1231950206*x^16 - 430200535*x^15 - 588270804*x^14 + 164470157*x^13 + 202516659*x^12 - 43750410*x^11 - 48240456*x^10 + 7822102*x^9 + 7480922*x^8 - 908640*x^7 - 688078*x^6 + 65581*x^5 + 31906*x^4 - 2504*x^3 - 504*x^2 + 24*x + 1, 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]])