\\ Pari/GP code for working with number field 43.43.444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961.1

\\ 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 - 210*y^41 + 177*y^40 + 19424*y^39 - 12392*y^38 - 1053196*y^37 + 410572*y^36 + 37567316*y^35 - 4029555*y^34 - 936601673*y^33 - 185951871*y^32 + 16894280750*y^31 + 8734312688*y^30 - 224649876009*y^29 - 189102381409*y^28 + 2218265633870*y^27 + 2604819705111*y^26 - 16218499586789*y^25 - 24791417254787*y^24 + 86521587673786*y^23 + 168116844068603*y^22 - 325504806681996*y^21 - 819692515707425*y^20 + 795734675389219*y^19 + 2859558999201013*y^18 - 937302717934781*y^17 - 7011360351157045*y^16 - 902655412698460*y^15 + 11681897146129224*y^14 + 5604720542659268*y^13 - 12486138055465655*y^12 - 9886946345787043*y^11 + 7728522779787490*y^10 + 9075115953805640*y^9 - 2183897418754987*y^8 - 4579013283800046*y^7 - 40649514655288*y^6 + 1249295360564226*y^5 + 160575454035341*y^4 - 166814479770181*y^3 - 29312399288701*y^2 + 7715035819499*y + 1403424452501, 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 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501, 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]])