\\ Pari/GP code for working with number field 42.0.121842012423466724043945276342536999203112328184628981033554149680639161638328200007.1

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

\\ Define the number field: 
K = bnfinit(y^42 - y^41 + 62*y^40 - 55*y^39 + 1885*y^38 - 1493*y^37 + 36890*y^36 - 26096*y^35 + 516884*y^34 - 325868*y^33 + 5474760*y^32 - 3063183*y^31 + 45214327*y^30 - 22319399*y^29 + 296463014*y^28 - 128129100*y^27 + 1558565095*y^26 - 584173775*y^25 + 6595257074*y^24 - 2118905803*y^23 + 22443263173*y^22 - 6093288301*y^21 + 61089146298*y^20 - 13772999112*y^19 + 131703790552*y^18 - 24134677936*y^17 + 221581028128*y^16 - 32118789120*y^15 + 284876509184*y^14 - 31567940608*y^13 + 271846162432*y^12 - 21964134400*y^11 + 184905250816*y^10 - 10296512512*y^9 + 84546772992*y^8 - 2848260096*y^7 + 23814799360*y^6 - 531300352*y^5 + 3554017280*y^4 + 28835840*y^3 + 213385216*y^2 - 11534336*y + 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 - x^41 + 62*x^40 - 55*x^39 + 1885*x^38 - 1493*x^37 + 36890*x^36 - 26096*x^35 + 516884*x^34 - 325868*x^33 + 5474760*x^32 - 3063183*x^31 + 45214327*x^30 - 22319399*x^29 + 296463014*x^28 - 128129100*x^27 + 1558565095*x^26 - 584173775*x^25 + 6595257074*x^24 - 2118905803*x^23 + 22443263173*x^22 - 6093288301*x^21 + 61089146298*x^20 - 13772999112*x^19 + 131703790552*x^18 - 24134677936*x^17 + 221581028128*x^16 - 32118789120*x^15 + 284876509184*x^14 - 31567940608*x^13 + 271846162432*x^12 - 21964134400*x^11 + 184905250816*x^10 - 10296512512*x^9 + 84546772992*x^8 - 2848260096*x^7 + 23814799360*x^6 - 531300352*x^5 + 3554017280*x^4 + 28835840*x^3 + 213385216*x^2 - 11534336*x + 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]])