\\ Pari/GP code for working with number field 42.0.409216671487706896433400972530400813921910324685198934110144125610159603082437813389667.1

\\ 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 + 20*y^40 - 25*y^39 - 8*y^38 - 138*y^37 - 2077*y^36 + 839*y^35 - 7305*y^34 + 8805*y^33 + 67596*y^32 - 12107*y^31 + 444232*y^30 - 141646*y^29 + 238560*y^28 + 2150502*y^27 - 3285732*y^26 + 14361861*y^25 - 708679*y^24 + 4004385*y^23 + 63920534*y^22 - 73366557*y^21 + 232544983*y^20 + 114300646*y^19 + 134380032*y^18 + 644451241*y^17 - 110366019*y^16 + 50739485*y^15 + 3000292231*y^14 - 3874326960*y^13 + 7531027885*y^12 - 11845394651*y^11 + 8115584321*y^10 - 10398559445*y^9 + 11442787794*y^8 - 3901741530*y^7 + 3575301015*y^6 - 3614499453*y^5 - 715331685*y^4 + 331663193*y^3 + 2507361126*y^2 - 1452650295*y + 649728353, 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 + 20*x^40 - 25*x^39 - 8*x^38 - 138*x^37 - 2077*x^36 + 839*x^35 - 7305*x^34 + 8805*x^33 + 67596*x^32 - 12107*x^31 + 444232*x^30 - 141646*x^29 + 238560*x^28 + 2150502*x^27 - 3285732*x^26 + 14361861*x^25 - 708679*x^24 + 4004385*x^23 + 63920534*x^22 - 73366557*x^21 + 232544983*x^20 + 114300646*x^19 + 134380032*x^18 + 644451241*x^17 - 110366019*x^16 + 50739485*x^15 + 3000292231*x^14 - 3874326960*x^13 + 7531027885*x^12 - 11845394651*x^11 + 8115584321*x^10 - 10398559445*x^9 + 11442787794*x^8 - 3901741530*x^7 + 3575301015*x^6 - 3614499453*x^5 - 715331685*x^4 + 331663193*x^3 + 2507361126*x^2 - 1452650295*x + 649728353, 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]])