\\ Pari/GP code for working with number field 44.0.6819629148478549071885414510662846662953220513498055755117351820034888330697796222976.1

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

\\ Define the number field: 
K = bnfinit(y^44 - 42*y^42 + 1004*y^40 - 16416*y^38 + 203056*y^36 - 1976896*y^34 + 15592256*y^32 - 100902784*y^30 + 540972800*y^28 - 2406430208*y^26 + 8885665792*y^24 - 27062865920*y^22 + 67552104448*y^20 - 136133230592*y^18 + 218558169088*y^16 - 271461023744*y^14 + 255644925952*y^12 - 171145035776*y^10 + 80371253248*y^8 - 21592276992*y^6 + 3817865216*y^4 - 138412032*y^2 + 4194304, 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 - 42*x^42 + 1004*x^40 - 16416*x^38 + 203056*x^36 - 1976896*x^34 + 15592256*x^32 - 100902784*x^30 + 540972800*x^28 - 2406430208*x^26 + 8885665792*x^24 - 27062865920*x^22 + 67552104448*x^20 - 136133230592*x^18 + 218558169088*x^16 - 271461023744*x^14 + 255644925952*x^12 - 171145035776*x^10 + 80371253248*x^8 - 21592276992*x^6 + 3817865216*x^4 - 138412032*x^2 + 4194304, 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]])