\\ Pari/GP code for working with number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.2

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

\\ Define the number field: 
K = bnfinit(y^44 - y^43 - 23*y^42 + 24*y^41 + 298*y^40 - 322*y^39 - 2644*y^38 + 2966*y^37 + 17733*y^36 - 20699*y^35 - 93680*y^34 + 114379*y^33 + 400546*y^32 - 514925*y^31 - 1402310*y^30 + 1917235*y^29 + 4042594*y^28 - 5959829*y^27 - 9550726*y^26 + 15510555*y^25 + 18239299*y^24 - 33749854*y^23 - 27151731*y^22 + 60965664*y^21 + 28718628*y^20 - 92631926*y^19 - 11478724*y^18 + 118848820*y^17 - 33434940*y^16 - 1406311*y^15 - 22927136*y^14 - 576983889*y^13 + 635212253*y^12 + 643308528*y^11 - 1294716116*y^10 + 1724346364*y^9 - 424183664*y^8 - 3407721010*y^7 + 3830694322*y^6 + 209294181*y^5 - 4039808114*y^4 + 4041269764*y^3 - 1473288*y^2 - 4104644424*y + 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 - x^43 - 23*x^42 + 24*x^41 + 298*x^40 - 322*x^39 - 2644*x^38 + 2966*x^37 + 17733*x^36 - 20699*x^35 - 93680*x^34 + 114379*x^33 + 400546*x^32 - 514925*x^31 - 1402310*x^30 + 1917235*x^29 + 4042594*x^28 - 5959829*x^27 - 9550726*x^26 + 15510555*x^25 + 18239299*x^24 - 33749854*x^23 - 27151731*x^22 + 60965664*x^21 + 28718628*x^20 - 92631926*x^19 - 11478724*x^18 + 118848820*x^17 - 33434940*x^16 - 1406311*x^15 - 22927136*x^14 - 576983889*x^13 + 635212253*x^12 + 643308528*x^11 - 1294716116*x^10 + 1724346364*x^9 - 424183664*x^8 - 3407721010*x^7 + 3830694322*x^6 + 209294181*x^5 - 4039808114*x^4 + 4041269764*x^3 - 1473288*x^2 - 4104644424*x + 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]])