\\ Pari/GP code for working with number field 32.0.2088443876129429457733048543333054873029337200425307489036314630977400864768.1

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

\\ Define the number field: 
K = bnfinit(y^32 + 416*y^30 + 78416*y^28 + 8858304*y^26 + 668327400*y^24 + 35525314240*y^22 + 1367724598240*y^20 + 38608911516032*y^18 + 799928385472788*y^16 + 12056891607126080*y^14 + 129666388829365024*y^12 + 963236031303854464*y^10 + 4695775652606290512*y^8 + 13840180870839593088*y^6 + 21419327538204132160*y^4 + 13103588611607233792*y^2 + 1330833218366359682, 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 + 416*x^30 + 78416*x^28 + 8858304*x^26 + 668327400*x^24 + 35525314240*x^22 + 1367724598240*x^20 + 38608911516032*x^18 + 799928385472788*x^16 + 12056891607126080*x^14 + 129666388829365024*x^12 + 963236031303854464*x^10 + 4695775652606290512*x^8 + 13840180870839593088*x^6 + 21419327538204132160*x^4 + 13103588611607233792*x^2 + 1330833218366359682, 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]])