\\ Pari/GP code for working with number field 42.0.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1

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

\\ Define the number field: 
K = bnfinit(y^42 + 84*y^40 + 3276*y^38 + 78736*y^36 + 1305360*y^34 + 15833664*y^32 + 145435136*y^30 + 1032886400*y^28 + 5741639680*y^26 + 25131904000*y^24 + 86707088384*y^22 + 234966480896*y^20 + 496154537984*y^18 + 805934137344*y^16 + 988445753344*y^14 + 891877195776*y^12 + 571258175488*y^10 + 247113187328*y^8 + 67166797824*y^6 + 10250354688*y^4 + 719323136*y^2 + 14680064, 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 + 84*x^40 + 3276*x^38 + 78736*x^36 + 1305360*x^34 + 15833664*x^32 + 145435136*x^30 + 1032886400*x^28 + 5741639680*x^26 + 25131904000*x^24 + 86707088384*x^22 + 234966480896*x^20 + 496154537984*x^18 + 805934137344*x^16 + 988445753344*x^14 + 891877195776*x^12 + 571258175488*x^10 + 247113187328*x^8 + 67166797824*x^6 + 10250354688*x^4 + 719323136*x^2 + 14680064, 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]])