\\ Pari/GP code for working with number field 44.0.220354102204022216469632262406301193826529556992340065310359890959262847900390625.1

\\ 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 + 32*y^42 - 21*y^41 + 608*y^40 - 302*y^39 + 7441*y^38 - 2334*y^37 + 66293*y^36 - 12644*y^35 + 432280*y^34 - 28161*y^33 + 2141316*y^32 + 33285*y^31 + 8001765*y^30 + 693375*y^29 + 23077394*y^28 + 2792911*y^27 + 51292654*y^26 + 7229970*y^25 + 89635724*y^24 + 12947356*y^23 + 123068074*y^22 + 17592284*y^21 + 133953248*y^20 + 17840654*y^19 + 114270996*y^18 + 13803865*y^17 + 76240835*y^16 + 7576064*y^15 + 38619954*y^14 + 2830121*y^13 + 14698338*y^12 + 489978*y^11 + 3921204*y^10 - 46901*y^9 + 736586*y^8 - 41505*y^7 + 78482*y^6 - 8849*y^5 + 5761*y^4 - 386*y^3 + 117*y^2 + 6*y + 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 - x^43 + 32*x^42 - 21*x^41 + 608*x^40 - 302*x^39 + 7441*x^38 - 2334*x^37 + 66293*x^36 - 12644*x^35 + 432280*x^34 - 28161*x^33 + 2141316*x^32 + 33285*x^31 + 8001765*x^30 + 693375*x^29 + 23077394*x^28 + 2792911*x^27 + 51292654*x^26 + 7229970*x^25 + 89635724*x^24 + 12947356*x^23 + 123068074*x^22 + 17592284*x^21 + 133953248*x^20 + 17840654*x^19 + 114270996*x^18 + 13803865*x^17 + 76240835*x^16 + 7576064*x^15 + 38619954*x^14 + 2830121*x^13 + 14698338*x^12 + 489978*x^11 + 3921204*x^10 - 46901*x^9 + 736586*x^8 - 41505*x^7 + 78482*x^6 - 8849*x^5 + 5761*x^4 - 386*x^3 + 117*x^2 + 6*x + 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]])