\\ Pari/GP code for working with number field 31.31.606473279060866185803408353837168008326030742006308575695635765007438922801.1

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

\\ Define the number field: 
K = bnfinit(y^31 - y^30 - 150*y^29 + 117*y^28 + 9434*y^27 - 4958*y^26 - 328880*y^25 + 78328*y^24 + 7098810*y^23 + 507925*y^22 - 100297691*y^21 - 39217806*y^20 + 950942678*y^19 + 677567200*y^18 - 6056859921*y^17 - 6293114670*y^16 + 25303767350*y^15 + 35315191500*y^14 - 65145555320*y^13 - 122001770825*y^12 + 87706314807*y^11 + 253609179978*y^10 - 19776440614*y^9 - 299693882286*y^8 - 98200795275*y^7 + 175359843704*y^6 + 113820457794*y^5 - 30260338943*y^4 - 38729335872*y^3 - 5811762482*y^2 + 1056847214*y - 26458109, 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^31 - x^30 - 150*x^29 + 117*x^28 + 9434*x^27 - 4958*x^26 - 328880*x^25 + 78328*x^24 + 7098810*x^23 + 507925*x^22 - 100297691*x^21 - 39217806*x^20 + 950942678*x^19 + 677567200*x^18 - 6056859921*x^17 - 6293114670*x^16 + 25303767350*x^15 + 35315191500*x^14 - 65145555320*x^13 - 122001770825*x^12 + 87706314807*x^11 + 253609179978*x^10 - 19776440614*x^9 - 299693882286*x^8 - 98200795275*x^7 + 175359843704*x^6 + 113820457794*x^5 - 30260338943*x^4 - 38729335872*x^3 - 5811762482*x^2 + 1056847214*x - 26458109, 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]])