\\ Pari/GP code for working with number field 42.0.1236405605609949863710440275882328463150065157474288679507165498832965910232619214817863.1

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

\\ Define the number field: 
K = bnfinit(y^42 + 6*y^40 - 4*y^39 - 207*y^38 - 72*y^37 - 857*y^36 + 225*y^35 + 13524*y^34 + 4498*y^33 + 47646*y^32 - 1779*y^31 - 319728*y^30 + 56511*y^29 - 873558*y^28 + 1268358*y^27 + 3182931*y^26 - 287550*y^25 + 12585242*y^24 - 25238916*y^23 + 37767705*y^22 + 35532417*y^21 + 14671665*y^20 + 387146580*y^19 - 33604015*y^18 + 83426931*y^17 + 893018691*y^16 - 978644267*y^15 + 2536898589*y^14 + 320653281*y^13 + 1627927205*y^12 + 1198131426*y^11 - 1595121042*y^10 - 2227675498*y^9 - 1701355491*y^8 - 2415610293*y^7 - 905477771*y^6 + 53305794*y^5 + 1490523864*y^4 + 2070850257*y^3 + 1799361396*y^2 + 662562267*y + 198529417, 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 + 6*x^40 - 4*x^39 - 207*x^38 - 72*x^37 - 857*x^36 + 225*x^35 + 13524*x^34 + 4498*x^33 + 47646*x^32 - 1779*x^31 - 319728*x^30 + 56511*x^29 - 873558*x^28 + 1268358*x^27 + 3182931*x^26 - 287550*x^25 + 12585242*x^24 - 25238916*x^23 + 37767705*x^22 + 35532417*x^21 + 14671665*x^20 + 387146580*x^19 - 33604015*x^18 + 83426931*x^17 + 893018691*x^16 - 978644267*x^15 + 2536898589*x^14 + 320653281*x^13 + 1627927205*x^12 + 1198131426*x^11 - 1595121042*x^10 - 2227675498*x^9 - 1701355491*x^8 - 2415610293*x^7 - 905477771*x^6 + 53305794*x^5 + 1490523864*x^4 + 2070850257*x^3 + 1799361396*x^2 + 662562267*x + 198529417, 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]])