\\ Pari/GP code for working with number field 44.0.21142110273569853339470849219791293286477383243143882449296109338159240255146818984969.1

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

\\ Define the number field: 
K = bnfinit(y^44 - y^43 - 4*y^42 + 9*y^41 + 11*y^40 - 56*y^39 + y^38 + 279*y^37 - 284*y^36 - 1111*y^35 + 2531*y^34 + 3024*y^33 - 15679*y^32 + 559*y^31 + 77836*y^30 - 80631*y^29 - 308549*y^28 + 711704*y^27 + 831041*y^26 - 4389561*y^25 + 234356*y^24 + 21713449*y^23 - 22885229*y^22 + 108567245*y^21 + 5858900*y^20 - 548695125*y^19 + 519400625*y^18 + 2224075000*y^17 - 4821078125*y^16 - 6299296875*y^15 + 30404687500*y^14 + 1091796875*y^13 - 153115234375*y^12 + 147656250000*y^11 + 617919921875*y^10 - 1356201171875*y^9 - 1733398437500*y^8 + 8514404296875*y^7 + 152587890625*y^6 - 42724609375000*y^5 + 41961669921875*y^4 + 171661376953125*y^3 - 381469726562500*y^2 - 476837158203125*y + 2384185791015625, 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 - 4*x^42 + 9*x^41 + 11*x^40 - 56*x^39 + x^38 + 279*x^37 - 284*x^36 - 1111*x^35 + 2531*x^34 + 3024*x^33 - 15679*x^32 + 559*x^31 + 77836*x^30 - 80631*x^29 - 308549*x^28 + 711704*x^27 + 831041*x^26 - 4389561*x^25 + 234356*x^24 + 21713449*x^23 - 22885229*x^22 + 108567245*x^21 + 5858900*x^20 - 548695125*x^19 + 519400625*x^18 + 2224075000*x^17 - 4821078125*x^16 - 6299296875*x^15 + 30404687500*x^14 + 1091796875*x^13 - 153115234375*x^12 + 147656250000*x^11 + 617919921875*x^10 - 1356201171875*x^9 - 1733398437500*x^8 + 8514404296875*x^7 + 152587890625*x^6 - 42724609375000*x^5 + 41961669921875*x^4 + 171661376953125*x^3 - 381469726562500*x^2 - 476837158203125*x + 2384185791015625, 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]])