\\ Pari/GP code for working with number field 32.0.5037920877776643425304576000000000000000000000000.1

\\ 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 + 3*y^30 - 2*y^29 - 12*y^27 + 17*y^26 - 24*y^25 + 3*y^24 + 8*y^23 + 101*y^22 - 128*y^21 + 208*y^20 - 34*y^19 + 39*y^18 - 726*y^17 + 877*y^16 - 1452*y^15 + 156*y^14 - 272*y^13 + 3328*y^12 - 4096*y^11 + 6464*y^10 + 1024*y^9 + 768*y^8 - 12288*y^7 + 17408*y^6 - 24576*y^5 - 16384*y^3 + 49152*y^2 - 65536*y + 65536, 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 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536, 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]])