\\ Pari/GP code for working with number field 42.42.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1

\\ 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 - 1032886144*y^28 + 5741625344*y^26 - 25131545600*y^24 + 86701812736*y^22 - 234915702784*y^20 + 495818960896*y^18 - 804378279936*y^16 + 983365402624*y^14 - 880319397888*y^12 + 553414557696*y^10 - 229269569536*y^8 + 56479711232*y^6 - 6910640128*y^4 + 308281344*y^2 - 2097152, 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 - 1032886144*x^28 + 5741625344*x^26 - 25131545600*x^24 + 86701812736*x^22 - 234915702784*x^20 + 495818960896*x^18 - 804378279936*x^16 + 983365402624*x^14 - 880319397888*x^12 + 553414557696*x^10 - 229269569536*x^8 + 56479711232*x^6 - 6910640128*x^4 + 308281344*x^2 - 2097152, 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]])