\\ Pari/GP code for working with number field 44.0.202576997637141672681589718254834044731913984308775018218427786812480361023243743657984.1

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

\\ Define the number field: 
K = bnfinit(y^44 - 63*y^42 + 1954*y^40 - 39159*y^38 + 564136*y^36 - 6171204*y^34 + 52905976*y^32 - 362220741*y^30 + 2002105655*y^28 - 8980281297*y^26 + 32710198942*y^24 - 96439370415*y^22 + 228524812948*y^20 - 430395212448*y^18 + 634098522688*y^16 - 714985491456*y^14 + 598979244032*y^12 - 357650202624*y^10 + 143735308288*y^8 - 35340484608*y^6 + 4755030016*y^4 - 207618048*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 - 63*x^42 + 1954*x^40 - 39159*x^38 + 564136*x^36 - 6171204*x^34 + 52905976*x^32 - 362220741*x^30 + 2002105655*x^28 - 8980281297*x^26 + 32710198942*x^24 - 96439370415*x^22 + 228524812948*x^20 - 430395212448*x^18 + 634098522688*x^16 - 714985491456*x^14 + 598979244032*x^12 - 357650202624*x^10 + 143735308288*x^8 - 35340484608*x^6 + 4755030016*x^4 - 207618048*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]])