\\ Pari/GP code for working with number field 44.44.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.1

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

\\ Define the number field: 
K = bnfinit(y^44 - 69*y^42 + 2139*y^40 - 39468*y^38 + 484495*y^36 - 4193682*y^34 + 26499197*y^32 - 125010267*y^30 + 447055439*y^28 - 1224773115*y^26 + 2588447416*y^24 - 4235761650*y^22 + 5369828745*y^20 - 5258413410*y^18 + 3950328660*y^16 - 2250421545*y^14 + 955257620*y^12 - 294507525*y^10 + 63543480*y^8 - 9077640*y^6 + 785565*y^4 - 34914*y^2 + 529, 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 - 69*x^42 + 2139*x^40 - 39468*x^38 + 484495*x^36 - 4193682*x^34 + 26499197*x^32 - 125010267*x^30 + 447055439*x^28 - 1224773115*x^26 + 2588447416*x^24 - 4235761650*x^22 + 5369828745*x^20 - 5258413410*x^18 + 3950328660*x^16 - 2250421545*x^14 + 955257620*x^12 - 294507525*x^10 + 63543480*x^8 - 9077640*x^6 + 785565*x^4 - 34914*x^2 + 529, 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]])