\\ Pari/GP code for working with number field 32.32.2318482735674757180308401186646582923236113207344277714859125196933746337890625.2

\\ 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 + 254*y^29 + 15176*y^28 - 22502*y^27 - 659185*y^26 + 1014228*y^25 + 17695139*y^24 - 27276312*y^23 - 311391664*y^22 + 475350824*y^21 + 3700993657*y^20 - 5615084090*y^19 - 30036394961*y^18 + 45970229245*y^17 + 165619980768*y^16 - 262233944896*y^15 - 606028450275*y^14 + 1031625598042*y^13 + 1394173531832*y^12 - 2722189078308*y^11 - 1767203506112*y^10 + 4578752971493*y^9 + 675907939625*y^8 - 4492953890543*y^7 + 922086899405*y^6 + 2162678325954*y^5 - 1005718811728*y^4 - 309785246453*y^3 + 244081450784*y^2 - 15465730875*y - 7522269649, 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 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1014228*x^25 + 17695139*x^24 - 27276312*x^23 - 311391664*x^22 + 475350824*x^21 + 3700993657*x^20 - 5615084090*x^19 - 30036394961*x^18 + 45970229245*x^17 + 165619980768*x^16 - 262233944896*x^15 - 606028450275*x^14 + 1031625598042*x^13 + 1394173531832*x^12 - 2722189078308*x^11 - 1767203506112*x^10 + 4578752971493*x^9 + 675907939625*x^8 - 4492953890543*x^7 + 922086899405*x^6 + 2162678325954*x^5 - 1005718811728*x^4 - 309785246453*x^3 + 244081450784*x^2 - 15465730875*x - 7522269649, 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]])