\\ Pari/GP code for working with number field 32.32.2318482735674757180308401186646582923236113207344277714859125196933746337890625.1

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

\\ Define the number field: 
K = bnfinit(y^32 - y^31 - 192*y^30 + 254*y^29 + 15176*y^28 - 22502*y^27 - 659185*y^26 + 1028778*y^25 + 17622874*y^24 - 28355437*y^23 - 306303044*y^22 + 508700394*y^21 + 3552738857*y^20 - 6168097580*y^19 - 27704655126*y^18 + 51265014015*y^17 + 143975452663*y^16 - 291596165966*y^15 - 484526195470*y^14 + 1120910038107*y^13 + 989059812617*y^12 - 2853145220348*y^11 - 1010392771187*y^10 + 4674809722638*y^9 - 46102330825*y^8 - 4692807095098*y^7 + 1274517894230*y^6 + 2597013247154*y^5 - 1244875523883*y^4 - 588593534703*y^3 + 420651733214*y^2 - 11758954445*y - 18277845959, 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 - x^31 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1028778*x^25 + 17622874*x^24 - 28355437*x^23 - 306303044*x^22 + 508700394*x^21 + 3552738857*x^20 - 6168097580*x^19 - 27704655126*x^18 + 51265014015*x^17 + 143975452663*x^16 - 291596165966*x^15 - 484526195470*x^14 + 1120910038107*x^13 + 989059812617*x^12 - 2853145220348*x^11 - 1010392771187*x^10 + 4674809722638*x^9 - 46102330825*x^8 - 4692807095098*x^7 + 1274517894230*x^6 + 2597013247154*x^5 - 1244875523883*x^4 - 588593534703*x^3 + 420651733214*x^2 - 11758954445*x - 18277845959, 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]])