\\ Pari/GP code for working with number field 44.0.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.1

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

\\ Define the number field: 
K = bnfinit(y^44 + 41*y^42 + 784*y^40 + 9282*y^38 + 76180*y^36 + 459888*y^34 + 2114697*y^32 + 7568133*y^30 + 21358299*y^28 + 47872465*y^26 + 85431991*y^24 + 121194085*y^22 + 135920335*y^20 + 119357605*y^18 + 80900650*y^16 + 41459620*y^14 + 15683335*y^12 + 3901015*y^10 + 2054320*y^8 - 6489250*y^6 + 32853580*y^4 - 164244624*y^2 + 821223649, 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 + 41*x^42 + 784*x^40 + 9282*x^38 + 76180*x^36 + 459888*x^34 + 2114697*x^32 + 7568133*x^30 + 21358299*x^28 + 47872465*x^26 + 85431991*x^24 + 121194085*x^22 + 135920335*x^20 + 119357605*x^18 + 80900650*x^16 + 41459620*x^14 + 15683335*x^12 + 3901015*x^10 + 2054320*x^8 - 6489250*x^6 + 32853580*x^4 - 164244624*x^2 + 821223649, 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]])