\\ Pari/GP code for working with number field 42.42.104017955712751803355033526522081856753017553948018605377854818344593048095703125.1

\\ 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 - 61*y^40 + 56*y^39 + 1654*y^38 - 1379*y^37 - 26380*y^36 + 19735*y^35 + 276302*y^34 - 183047*y^33 - 2012061*y^32 + 1164081*y^31 + 10536175*y^30 - 5248100*y^29 - 40536685*y^28 + 17139885*y^27 + 116229751*y^26 - 41141876*y^25 - 250648402*y^24 + 73243022*y^23 + 408521488*y^22 - 97102633*y^21 - 503598537*y^20 + 95768202*y^19 + 467783098*y^18 - 69790048*y^17 - 324502952*y^16 + 37071837*y^15 + 165565463*y^14 - 14037193*y^13 - 60679502*y^12 + 3661307*y^11 + 15419833*y^10 - 625768*y^9 - 2575782*y^8 + 66132*y^7 + 260062*y^6 - 4477*y^5 - 13750*y^4 + 275*y^3 + 286*y^2 - 11*y - 1, 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 - 61*x^40 + 56*x^39 + 1654*x^38 - 1379*x^37 - 26380*x^36 + 19735*x^35 + 276302*x^34 - 183047*x^33 - 2012061*x^32 + 1164081*x^31 + 10536175*x^30 - 5248100*x^29 - 40536685*x^28 + 17139885*x^27 + 116229751*x^26 - 41141876*x^25 - 250648402*x^24 + 73243022*x^23 + 408521488*x^22 - 97102633*x^21 - 503598537*x^20 + 95768202*x^19 + 467783098*x^18 - 69790048*x^17 - 324502952*x^16 + 37071837*x^15 + 165565463*x^14 - 14037193*x^13 - 60679502*x^12 + 3661307*x^11 + 15419833*x^10 - 625768*x^9 - 2575782*x^8 + 66132*x^7 + 260062*x^6 - 4477*x^5 - 13750*x^4 + 275*x^3 + 286*x^2 - 11*x - 1, 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]])