\\ Pari/GP code for working with number field 42.0.2011999877834826766225008958075022926316813554075780070378415668274435623250777079808.1

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

\\ Define the number field: 
K = bnfinit(y^42 + 82*y^40 + 3120*y^38 + 73112*y^36 + 1181040*y^34 + 13948704*y^32 + 124658688*y^30 + 860738560*y^28 + 4647988224*y^26 + 19746355200*y^24 + 66060533760*y^22 + 173408901120*y^20 + 354276249600*y^18 + 555941191680*y^16 + 657270374400*y^14 + 569634324480*y^12 + 348109864960*y^10 + 141764198400*y^8 + 35283533824*y^6 + 4642570240*y^4 + 242221056*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 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*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]])