\\ Pari/GP code for working with number field 44.0.116567320065927752512435466812933331534234135648947894549180382317450046539306640625.1

\\ 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 + 43*y^42 - 39*y^41 + 856*y^40 - 700*y^39 + 10478*y^38 - 7678*y^37 + 88356*y^36 - 57644*y^35 + 545008*y^34 - 314432*y^33 + 2548537*y^32 - 1290809*y^31 + 9238279*y^30 - 4075043*y^29 + 26320891*y^28 - 10020719*y^27 + 59401115*y^26 - 19318239*y^25 + 106508095*y^24 - 29235139*y^23 + 151556799*y^22 - 34552164*y^21 + 170215728*y^20 - 30533255*y^19 + 148629623*y^18 - 11758433*y^17 + 94928616*y^16 + 36112685*y^15 + 30135721*y^14 + 126072207*y^13 - 23633656*y^12 + 220881720*y^11 - 44029406*y^10 + 233441302*y^9 - 35764283*y^8 + 155778230*y^7 - 8938748*y^6 + 18093321*y^5 + 18069001*y^4 + 120313546*y^3 - 192566274*y^2 - 256263936*y + 1026529561, 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 + 43*x^42 - 39*x^41 + 856*x^40 - 700*x^39 + 10478*x^38 - 7678*x^37 + 88356*x^36 - 57644*x^35 + 545008*x^34 - 314432*x^33 + 2548537*x^32 - 1290809*x^31 + 9238279*x^30 - 4075043*x^29 + 26320891*x^28 - 10020719*x^27 + 59401115*x^26 - 19318239*x^25 + 106508095*x^24 - 29235139*x^23 + 151556799*x^22 - 34552164*x^21 + 170215728*x^20 - 30533255*x^19 + 148629623*x^18 - 11758433*x^17 + 94928616*x^16 + 36112685*x^15 + 30135721*x^14 + 126072207*x^13 - 23633656*x^12 + 220881720*x^11 - 44029406*x^10 + 233441302*x^9 - 35764283*x^8 + 155778230*x^7 - 8938748*x^6 + 18093321*x^5 + 18069001*x^4 + 120313546*x^3 - 192566274*x^2 - 256263936*x + 1026529561, 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]])