\\ Pari/GP code for working with number field 44.0.1555282614282872031686695202841759550456839851168205668482420621934906547732043786064001.1

\\ 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 - y^42 - 82*y^41 + 77*y^40 + 73*y^39 + 2795*y^38 - 2452*y^37 - 2193*y^36 - 51847*y^35 + 42213*y^34 + 35713*y^33 + 576416*y^32 - 432884*y^31 - 345487*y^30 - 3995639*y^29 + 2790104*y^28 + 2010938*y^27 + 17436469*y^26 - 11948291*y^25 - 6590980*y^24 - 47156450*y^23 + 36319435*y^22 + 9637447*y^21 + 76421426*y^20 - 81395780*y^19 + 4175237*y^18 - 78460015*y^17 + 124870243*y^16 - 45435254*y^15 + 82436729*y^14 - 106904257*y^13 + 95684734*y^12 - 78566744*y^11 + 62868338*y^10 - 89632848*y^9 + 65796469*y^8 - 36457247*y^7 + 9020998*y^6 - 24902176*y^5 + 25739321*y^4 + 13163316*y^3 - 5336842*y^2 + 199578*y + 1151329, 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 - x^42 - 82*x^41 + 77*x^40 + 73*x^39 + 2795*x^38 - 2452*x^37 - 2193*x^36 - 51847*x^35 + 42213*x^34 + 35713*x^33 + 576416*x^32 - 432884*x^31 - 345487*x^30 - 3995639*x^29 + 2790104*x^28 + 2010938*x^27 + 17436469*x^26 - 11948291*x^25 - 6590980*x^24 - 47156450*x^23 + 36319435*x^22 + 9637447*x^21 + 76421426*x^20 - 81395780*x^19 + 4175237*x^18 - 78460015*x^17 + 124870243*x^16 - 45435254*x^15 + 82436729*x^14 - 106904257*x^13 + 95684734*x^12 - 78566744*x^11 + 62868338*x^10 - 89632848*x^9 + 65796469*x^8 - 36457247*x^7 + 9020998*x^6 - 24902176*x^5 + 25739321*x^4 + 13163316*x^3 - 5336842*x^2 + 199578*x + 1151329, 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]])