\\ Pari/GP code for working with number field 43.43.9952594992767664919302480397055915291864099597331865326873171797085952964991380209309055259529.1

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

\\ Define the number field: 
K = bnfinit(y^43 - y^42 - 84*y^41 + 79*y^40 + 3160*y^39 - 2786*y^38 - 70521*y^37 + 58076*y^36 + 1042773*y^35 - 798942*y^34 - 10810701*y^33 + 7671648*y^32 + 81126975*y^31 - 53056499*y^30 - 448758019*y^29 + 268953093*y^28 + 1846875875*y^27 - 1007873658*y^26 - 5671315301*y^25 + 2797394685*y^24 + 12962901258*y^23 - 5730420663*y^22 - 21895326590*y^21 + 8590544711*y^20 + 27001558938*y^19 - 9297003415*y^18 - 23896748020*y^17 + 7121607714*y^16 + 14834334408*y^15 - 3755010451*y^14 - 6269987218*y^13 + 1310830451*y^12 + 1732796449*y^11 - 287529242*y^10 - 294221159*y^9 + 37041429*y^8 + 27505221*y^7 - 2599666*y^6 - 1134285*y^5 + 90620*y^4 + 12893*y^3 - 370*y^2 - 54*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^43 - x^42 - 84*x^41 + 79*x^40 + 3160*x^39 - 2786*x^38 - 70521*x^37 + 58076*x^36 + 1042773*x^35 - 798942*x^34 - 10810701*x^33 + 7671648*x^32 + 81126975*x^31 - 53056499*x^30 - 448758019*x^29 + 268953093*x^28 + 1846875875*x^27 - 1007873658*x^26 - 5671315301*x^25 + 2797394685*x^24 + 12962901258*x^23 - 5730420663*x^22 - 21895326590*x^21 + 8590544711*x^20 + 27001558938*x^19 - 9297003415*x^18 - 23896748020*x^17 + 7121607714*x^16 + 14834334408*x^15 - 3755010451*x^14 - 6269987218*x^13 + 1310830451*x^12 + 1732796449*x^11 - 287529242*x^10 - 294221159*x^9 + 37041429*x^8 + 27505221*x^7 - 2599666*x^6 - 1134285*x^5 + 90620*x^4 + 12893*x^3 - 370*x^2 - 54*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]])