\\ Pari/GP code for working with number field 44.44.191158492466532603992882058813386761401107535956984758058028173625115327713733756218489.1

\\ 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 - 86*y^42 + 81*y^41 + 3445*y^40 - 3040*y^39 - 85351*y^38 + 70151*y^37 + 1464534*y^36 - 1113779*y^35 - 18468767*y^34 + 12899872*y^33 + 177261061*y^32 - 112761701*y^31 - 1323060098*y^30 + 759251593*y^29 + 7780382701*y^28 - 3984124736*y^27 - 36301909279*y^26 + 16381285599*y^25 + 134698547590*y^24 - 52792119595*y^23 - 396744216375*y^22 + 132783613203*y^21 + 922279857882*y^20 - 258361552805*y^19 - 1675313579545*y^18 + 383501034280*y^17 + 2343182319195*y^16 - 425622641659*y^15 - 2471777086526*y^14 + 343273729047*y^13 + 1911152162483*y^12 - 192962821056*y^11 - 1040820630209*y^10 + 70437336577*y^9 + 377125555258*y^8 - 13999395253*y^7 - 83413545257*y^6 + 289196448*y^5 + 9759539347*y^4 + 438024589*y^3 - 453637806*y^2 - 49033617*y - 368597, 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 - 86*x^42 + 81*x^41 + 3445*x^40 - 3040*x^39 - 85351*x^38 + 70151*x^37 + 1464534*x^36 - 1113779*x^35 - 18468767*x^34 + 12899872*x^33 + 177261061*x^32 - 112761701*x^31 - 1323060098*x^30 + 759251593*x^29 + 7780382701*x^28 - 3984124736*x^27 - 36301909279*x^26 + 16381285599*x^25 + 134698547590*x^24 - 52792119595*x^23 - 396744216375*x^22 + 132783613203*x^21 + 922279857882*x^20 - 258361552805*x^19 - 1675313579545*x^18 + 383501034280*x^17 + 2343182319195*x^16 - 425622641659*x^15 - 2471777086526*x^14 + 343273729047*x^13 + 1911152162483*x^12 - 192962821056*x^11 - 1040820630209*x^10 + 70437336577*x^9 + 377125555258*x^8 - 13999395253*x^7 - 83413545257*x^6 + 289196448*x^5 + 9759539347*x^4 + 438024589*x^3 - 453637806*x^2 - 49033617*x - 368597, 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]])