\\ Pari/GP code for working with number field 43.43.101554487341926844216969846221080781237895685553160643066505060271306560421136844000042773102578137815468250505123323781432009.1

\\ 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 - 462*y^41 + 301*y^40 + 94630*y^39 - 49892*y^38 - 11463758*y^37 + 6368083*y^36 + 921810336*y^35 - 636827636*y^34 - 52222121579*y^33 + 47086072273*y^32 + 2155113502777*y^31 - 2510570612339*y^30 - 65969924993323*y^29 + 96629348183472*y^28 + 1509193690295722*y^27 - 2705517116526302*y^26 - 25775660913572833*y^25 + 55430418074111539*y^24 + 325598895562519548*y^23 - 832294062829303477*y^22 - 2983438147701667253*y^21 + 9129779342869086290*y^20 + 19125104314160845818*y^19 - 72594260758345520281*y^18 - 79441906767691419891*y^17 + 413102772817597404709*y^16 + 166866840849449475306*y^15 - 1650628224018564426381*y^14 + 139804498689687778120*y^13 + 4502348802434562518439*y^12 - 1981830512581480780535*y^11 - 8037383143222291539220*y^10 + 5701308150866478483031*y^9 + 8799946070456643867879*y^8 - 8188607772821299311768*y^7 - 5304980067316890480275*y^6 + 6100770874409784770241*y^5 + 1408273844297690430983*y^4 - 2063406363097082700573*y^3 - 90731765675438007935*y^2 + 193966087122261586736*y + 18221685305593614631, 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 - 462*x^41 + 301*x^40 + 94630*x^39 - 49892*x^38 - 11463758*x^37 + 6368083*x^36 + 921810336*x^35 - 636827636*x^34 - 52222121579*x^33 + 47086072273*x^32 + 2155113502777*x^31 - 2510570612339*x^30 - 65969924993323*x^29 + 96629348183472*x^28 + 1509193690295722*x^27 - 2705517116526302*x^26 - 25775660913572833*x^25 + 55430418074111539*x^24 + 325598895562519548*x^23 - 832294062829303477*x^22 - 2983438147701667253*x^21 + 9129779342869086290*x^20 + 19125104314160845818*x^19 - 72594260758345520281*x^18 - 79441906767691419891*x^17 + 413102772817597404709*x^16 + 166866840849449475306*x^15 - 1650628224018564426381*x^14 + 139804498689687778120*x^13 + 4502348802434562518439*x^12 - 1981830512581480780535*x^11 - 8037383143222291539220*x^10 + 5701308150866478483031*x^9 + 8799946070456643867879*x^8 - 8188607772821299311768*x^7 - 5304980067316890480275*x^6 + 6100770874409784770241*x^5 + 1408273844297690430983*x^4 - 2063406363097082700573*x^3 - 90731765675438007935*x^2 + 193966087122261586736*x + 18221685305593614631, 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]])