\\ Pari/GP code for working with number field 44.44.181375764426442332776050749828434842919201892356144879261148162186145782470703125.1

\\ 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 - 44*y^42 + 43*y^41 + 902*y^40 - 859*y^39 - 11441*y^38 + 10582*y^37 + 100567*y^36 - 89985*y^35 - 650236*y^34 + 560251*y^33 + 3203650*y^32 - 2643399*y^31 - 12294569*y^30 + 9651170*y^29 + 37253225*y^28 - 27602055*y^27 - 89808680*y^26 + 62206625*y^25 + 172779011*y^24 - 110572386*y^23 - 265002643*y^22 + 154430258*y^21 + 322457859*y^20 - 168027624*y^19 - 308473797*y^18 + 140446403*y^17 + 228768475*y^16 - 88323383*y^15 - 128871012*y^14 + 40552321*y^13 + 53558102*y^12 - 13016729*y^11 - 15742859*y^10 + 2742874*y^9 + 3073662*y^8 - 347233*y^7 - 361455*y^6 + 24089*y^5 + 21786*y^4 - 986*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 - 44*x^42 + 43*x^41 + 902*x^40 - 859*x^39 - 11441*x^38 + 10582*x^37 + 100567*x^36 - 89985*x^35 - 650236*x^34 + 560251*x^33 + 3203650*x^32 - 2643399*x^31 - 12294569*x^30 + 9651170*x^29 + 37253225*x^28 - 27602055*x^27 - 89808680*x^26 + 62206625*x^25 + 172779011*x^24 - 110572386*x^23 - 265002643*x^22 + 154430258*x^21 + 322457859*x^20 - 168027624*x^19 - 308473797*x^18 + 140446403*x^17 + 228768475*x^16 - 88323383*x^15 - 128871012*x^14 + 40552321*x^13 + 53558102*x^12 - 13016729*x^11 - 15742859*x^10 + 2742874*x^9 + 3073662*x^8 - 347233*x^7 - 361455*x^6 + 24089*x^5 + 21786*x^4 - 986*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]])