\\ Pari/GP code for working with number field 32.0.42655280577413385127292283461599409580230712890625.1

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

\\ Define the number field: 
K = bnfinit(y^32 - 2*y^31 - 14*y^30 + 32*y^29 + 110*y^28 - 256*y^27 - 822*y^26 + 1761*y^25 + 5076*y^24 - 6234*y^23 - 33617*y^22 + 7049*y^21 + 145847*y^20 + 80587*y^19 - 200297*y^18 - 1040496*y^17 + 909785*y^16 - 126545*y^15 + 4807173*y^14 - 6468687*y^13 + 4092091*y^12 - 12326083*y^11 + 15573178*y^10 - 9966741*y^9 + 15430857*y^8 - 15762104*y^7 + 7858456*y^6 - 7358464*y^5 + 5444944*y^4 - 823424*y^3 + 88384*y^2 - 5760*y + 256, 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 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256, 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]])