\\ Pari/GP code for working with number field 32.0.1156745857256138299057661301456078800598048178176.1

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

\\ Define the number field: 
K = bnfinit(y^32 + 10*y^30 - 28*y^29 + 56*y^28 - 208*y^27 + 604*y^26 - 876*y^25 + 2547*y^24 - 5760*y^23 + 7684*y^22 - 11384*y^21 + 37676*y^20 - 21776*y^19 + 12262*y^18 - 135456*y^17 + 96553*y^16 + 443172*y^15 + 1107228*y^14 + 758160*y^13 - 662400*y^12 - 2309040*y^11 - 1681344*y^10 + 1407456*y^9 + 4596912*y^8 + 3872448*y^7 - 62208*y^6 - 4385664*y^5 - 3545856*y^4 + 3919104*y^2 + 3359232*y + 1679616, 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 + 10*x^30 - 28*x^29 + 56*x^28 - 208*x^27 + 604*x^26 - 876*x^25 + 2547*x^24 - 5760*x^23 + 7684*x^22 - 11384*x^21 + 37676*x^20 - 21776*x^19 + 12262*x^18 - 135456*x^17 + 96553*x^16 + 443172*x^15 + 1107228*x^14 + 758160*x^13 - 662400*x^12 - 2309040*x^11 - 1681344*x^10 + 1407456*x^9 + 4596912*x^8 + 3872448*x^7 - 62208*x^6 - 4385664*x^5 - 3545856*x^4 + 3919104*x^2 + 3359232*x + 1679616, 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]])