\\ Pari/GP code for working with number field 42.42.86515994746897550947675385197225985831622982825258543026271873735800731799783414431744.1

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

\\ Define the number field: 
K = bnfinit(y^42 - 86*y^40 + 3440*y^38 - 84968*y^36 + 1450992*y^34 - 18175584*y^32 + 172913664*y^30 - 1276267520*y^28 + 7402351616*y^26 - 33963730944*y^24 + 123504476160*y^22 - 355075368960*y^20 + 801783091200*y^18 - 1406204190720*y^16 + 1884175073280*y^14 - 1884175073280*y^12 + 1360793108480*y^10 - 677317836800*y^8 + 216741707776*y^6 - 39926104064*y^4 + 3471835136*y^2 - 90177536, 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 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536, 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]])