\\ Pari/GP code for working with number field 44.44.482179487665033966874817964307376476160282778171317425736173928285756467369658548224.1

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

\\ Define the number field: 
K = bnfinit(y^44 - 44*y^42 + 901*y^40 - 11400*y^38 + 99790*y^36 - 641208*y^34 + 3131721*y^32 - 11878176*y^30 + 35442612*y^28 - 83778736*y^26 + 157236844*y^24 - 233880352*y^22 + 274130056*y^20 - 250699168*y^18 + 176290339*y^16 - 93382192*y^14 + 36217051*y^12 - 9883588*y^10 + 1792219*y^8 - 197912*y^6 + 11506*y^4 - 264*y^2 + 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 - 44*x^42 + 901*x^40 - 11400*x^38 + 99790*x^36 - 641208*x^34 + 3131721*x^32 - 11878176*x^30 + 35442612*x^28 - 83778736*x^26 + 157236844*x^24 - 233880352*x^22 + 274130056*x^20 - 250699168*x^18 + 176290339*x^16 - 93382192*x^14 + 36217051*x^12 - 9883588*x^10 + 1792219*x^8 - 197912*x^6 + 11506*x^4 - 264*x^2 + 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]])