\\ Pari/GP code for working with number field 42.42.496897759422042196258605771077406782550407598249513303021389442457964675897236469.1

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

\\ Define the number field: 
K = bnfinit(y^42 - y^41 - 67*y^40 + 62*y^39 + 1993*y^38 - 1704*y^37 - 34847*y^36 + 27519*y^35 + 399768*y^34 - 291838*y^33 - 3181854*y^32 + 2151119*y^31 + 18125561*y^30 - 11376768*y^29 - 75199314*y^28 + 43948028*y^27 + 229267037*y^26 - 125109956*y^25 - 515411960*y^24 + 263200585*y^23 + 854114535*y^22 - 408541162*y^21 - 1040280967*y^20 + 465602383*y^19 + 926307827*y^18 - 386475878*y^17 - 597992464*y^16 + 230741151*y^15 + 276295754*y^14 - 97190872*y^13 - 89602671*y^12 + 28039534*y^11 + 19807180*y^10 - 5297092*y^9 - 2853616*y^8 + 611802*y^7 + 248962*y^6 - 38836*y^5 - 11508*y^4 + 1155*y^3 + 217*y^2 - 14*y - 1, 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^42 - x^41 - 67*x^40 + 62*x^39 + 1993*x^38 - 1704*x^37 - 34847*x^36 + 27519*x^35 + 399768*x^34 - 291838*x^33 - 3181854*x^32 + 2151119*x^31 + 18125561*x^30 - 11376768*x^29 - 75199314*x^28 + 43948028*x^27 + 229267037*x^26 - 125109956*x^25 - 515411960*x^24 + 263200585*x^23 + 854114535*x^22 - 408541162*x^21 - 1040280967*x^20 + 465602383*x^19 + 926307827*x^18 - 386475878*x^17 - 597992464*x^16 + 230741151*x^15 + 276295754*x^14 - 97190872*x^13 - 89602671*x^12 + 28039534*x^11 + 19807180*x^10 - 5297092*x^9 - 2853616*x^8 + 611802*x^7 + 248962*x^6 - 38836*x^5 - 11508*x^4 + 1155*x^3 + 217*x^2 - 14*x - 1, 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]])