\\ Pari/GP code for working with number field 32.0.785411287106652838033363851140085365286337539393649349536153620008513161719185408.1

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

\\ Define the number field: 
K = bnfinit(y^32 + 928*y^30 + 390224*y^28 + 98336448*y^26 + 16550375400*y^24 + 1962506736320*y^22 + 168549136238560*y^20 + 10613779893422464*y^18 + 490555639449119508*y^16 + 16494044688724018240*y^14 + 395707126668569855776*y^12 + 6557432384793443324288*y^10 + 71312077184628696151632*y^8 + 468869689667356285628544*y^6 + 1618716785756349081336640*y^4 + 2209072319385135216882944*y^2 + 500492947360694697575042, 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 + 928*x^30 + 390224*x^28 + 98336448*x^26 + 16550375400*x^24 + 1962506736320*x^22 + 168549136238560*x^20 + 10613779893422464*x^18 + 490555639449119508*x^16 + 16494044688724018240*x^14 + 395707126668569855776*x^12 + 6557432384793443324288*x^10 + 71312077184628696151632*x^8 + 468869689667356285628544*x^6 + 1618716785756349081336640*x^4 + 2209072319385135216882944*x^2 + 500492947360694697575042, 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]])