\\ Pari/GP code for working with number field 44.0.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.2

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

\\ Define the number field: 
K = bnfinit(y^44 - 47*y^42 + 1036*y^40 - 14238*y^38 + 136828*y^36 - 977616*y^34 + 5391753*y^32 - 23533371*y^30 + 82732335*y^28 - 237396295*y^26 + 562190611*y^24 - 1110544651*y^22 + 1852435411*y^20 - 2651394691*y^18 + 3329083366*y^16 - 3775046236*y^14 + 3998027671*y^12 - 4080345361*y^10 + 4101904756*y^8 - 4105687106*y^6 + 4106094436*y^4 - 4106117712*y^2 + 4106118241, 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 - 47*x^42 + 1036*x^40 - 14238*x^38 + 136828*x^36 - 977616*x^34 + 5391753*x^32 - 23533371*x^30 + 82732335*x^28 - 237396295*x^26 + 562190611*x^24 - 1110544651*x^22 + 1852435411*x^20 - 2651394691*x^18 + 3329083366*x^16 - 3775046236*x^14 + 3998027671*x^12 - 4080345361*x^10 + 4101904756*x^8 - 4105687106*x^6 + 4106094436*x^4 - 4106117712*x^2 + 4106118241, 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]])