\\ Pari/GP code for working with number field 31.31.10780058321499447579019490584061143592974328542534512659762928215062178274195635731849.1

\\ 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 - 330*y^29 + 852*y^28 + 43784*y^27 - 173944*y^26 - 2923572*y^25 + 15562316*y^24 + 100635650*y^23 - 707709554*y^22 - 1616621784*y^21 + 17266725972*y^20 + 7046247776*y^19 - 243865756432*y^18 + 142454592564*y^17 + 2084163747748*y^16 - 2521472241195*y^15 - 10836712986085*y^14 + 18499228115810*y^13 + 32698257976552*y^12 - 74538290301632*y^11 - 47884603550112*y^10 + 168256221920320*y^9 + 4878741111552*y^8 - 195294436805376*y^7 + 62882730299136*y^6 + 92885896276480*y^5 - 43440624908288*y^4 - 15171034030080*y^3 + 6823763935232*y^2 + 786648006656*y + 5192548352, 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 - 330*x^29 + 852*x^28 + 43784*x^27 - 173944*x^26 - 2923572*x^25 + 15562316*x^24 + 100635650*x^23 - 707709554*x^22 - 1616621784*x^21 + 17266725972*x^20 + 7046247776*x^19 - 243865756432*x^18 + 142454592564*x^17 + 2084163747748*x^16 - 2521472241195*x^15 - 10836712986085*x^14 + 18499228115810*x^13 + 32698257976552*x^12 - 74538290301632*x^11 - 47884603550112*x^10 + 168256221920320*x^9 + 4878741111552*x^8 - 195294436805376*x^7 + 62882730299136*x^6 + 92885896276480*x^5 - 43440624908288*x^4 - 15171034030080*x^3 + 6823763935232*x^2 + 786648006656*x + 5192548352, 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]])