\\ Pari/GP code for working with number field 44.0.860115008245742907292219227824365111518443501386754869093198564719785672888549376.2

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

\\ Define the number field: 
K = bnfinit(y^44 + 23*y^42 + 299*y^40 + 2668*y^38 + 18055*y^36 + 96646*y^34 + 421245*y^32 + 1516689*y^30 + 4557519*y^28 + 11467961*y^26 + 24199128*y^24 + 42662286*y^22 + 62532561*y^20 + 75392022*y^18 + 73935156*y^16 + 57768387*y^14 + 35301228*y^12 + 16195335*y^10 + 5446584*y^8 + 1210352*y^6 + 180389*y^4 + 11638*y^2 + 529, 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 + 23*x^42 + 299*x^40 + 2668*x^38 + 18055*x^36 + 96646*x^34 + 421245*x^32 + 1516689*x^30 + 4557519*x^28 + 11467961*x^26 + 24199128*x^24 + 42662286*x^22 + 62532561*x^20 + 75392022*x^18 + 73935156*x^16 + 57768387*x^14 + 35301228*x^12 + 16195335*x^10 + 5446584*x^8 + 1210352*x^6 + 180389*x^4 + 11638*x^2 + 529, 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]])