\\ Pari/GP code for working with number field 42.42.2011999877834826766225008958075022926316813554075780070378415668274435623250777079808.1

\\ (Note that not all these functions may be available, and some may take a long time to execute.)

\\ Define the number field: 
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)

\\ 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

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
p = 7;  \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
idealfactors = idealprimedec(K, p); \\ get the data
vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])