\\ Pari/GP code for working with number field 32.32.1292684086480957274390235903780365469718835315185422412744961998012838206753.1

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

\\ Define the number field: 
K = bnfinit(y^32 - y^31 - 192*y^30 - 134*y^29 + 15855*y^28 + 35407*y^27 - 704581*y^26 - 2636561*y^25 + 17183464*y^24 + 98480308*y^23 - 188031041*y^22 - 2067532985*y^21 - 767939030*y^20 + 24362757568*y^19 + 47464737498*y^18 - 139397040512*y^17 - 559716324137*y^16 + 51715582409*y^15 + 2957237171335*y^14 + 3668327522541*y^13 - 5872939632229*y^12 - 17489038973421*y^11 - 5531030792826*y^10 + 26831580182190*y^9 + 33098017056437*y^8 - 1991967071353*y^7 - 29363649724089*y^6 - 18650067236811*y^5 + 1413749921964*y^4 + 5776725400421*y^3 + 2097291371636*y^2 + 190504344411*y - 14559956239, 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^32 - x^31 - 192*x^30 - 134*x^29 + 15855*x^28 + 35407*x^27 - 704581*x^26 - 2636561*x^25 + 17183464*x^24 + 98480308*x^23 - 188031041*x^22 - 2067532985*x^21 - 767939030*x^20 + 24362757568*x^19 + 47464737498*x^18 - 139397040512*x^17 - 559716324137*x^16 + 51715582409*x^15 + 2957237171335*x^14 + 3668327522541*x^13 - 5872939632229*x^12 - 17489038973421*x^11 - 5531030792826*x^10 + 26831580182190*x^9 + 33098017056437*x^8 - 1991967071353*x^7 - 29363649724089*x^6 - 18650067236811*x^5 + 1413749921964*x^4 + 5776725400421*x^3 + 2097291371636*x^2 + 190504344411*x - 14559956239, 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]])