\\ Pari/GP code for working with number field 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1

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

\\ Define the number field: 
K = bnfinit(y^42 - 63*y^40 + 1764*y^38 - 29043*y^36 - 29*y^35 + 313845*y^34 + 1295*y^33 - 2356263*y^32 - 24605*y^31 + 12710649*y^30 + 262360*y^29 - 50338309*y^28 - 1751015*y^27 + 148468453*y^26 + 7757582*y^25 - 329114310*y^24 - 23675820*y^23 + 550880022*y^22 + 50961633*y^21 - 696364921*y^20 - 78352652*y^19 + 661675518*y^18 + 86284422*y^17 - 467756891*y^16 - 67555180*y^15 + 241883075*y^14 + 36893815*y^13 - 89215119*y^12 - 13584956*y^11 + 22638112*y^10 + 3191321*y^9 - 3752308*y^8 - 437383*y^7 + 374801*y^6 + 29995*y^5 - 19355*y^4 - 798*y^3 + 343*y^2 + 14*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 - 63*x^40 + 1764*x^38 - 29043*x^36 - 29*x^35 + 313845*x^34 + 1295*x^33 - 2356263*x^32 - 24605*x^31 + 12710649*x^30 + 262360*x^29 - 50338309*x^28 - 1751015*x^27 + 148468453*x^26 + 7757582*x^25 - 329114310*x^24 - 23675820*x^23 + 550880022*x^22 + 50961633*x^21 - 696364921*x^20 - 78352652*x^19 + 661675518*x^18 + 86284422*x^17 - 467756891*x^16 - 67555180*x^15 + 241883075*x^14 + 36893815*x^13 - 89215119*x^12 - 13584956*x^11 + 22638112*x^10 + 3191321*x^9 - 3752308*x^8 - 437383*x^7 + 374801*x^6 + 29995*x^5 - 19355*x^4 - 798*x^3 + 343*x^2 + 14*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]])