\\ Pari/GP code for working with number field 42.42.776706980099412683270456983716625307007021218630434418395046564476510573119136503693312.1

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

\\ Define the number field: 
K = bnfinit(y^42 - 105*y^40 + 4977*y^38 - 140819*y^36 - 2*y^35 + 2650095*y^34 + 28*y^33 - 34991649*y^32 + 3416*y^31 + 333300632*y^30 - 147910*y^29 - 2319579309*y^28 + 2698850*y^27 + 11820066804*y^26 - 26976474*y^25 - 43812582639*y^24 + 153194076*y^23 + 116368453593*y^22 - 445815310*y^21 - 216225058245*y^20 + 272316800*y^19 + 271556250546*y^18 + 1986860386*y^17 - 220004406804*y^16 - 5445985720*y^15 + 108239023559*y^14 + 5365080098*y^13 - 29954210552*y^12 - 2176838734*y^11 + 4303639312*y^10 + 340937478*y^9 - 323375283*y^8 - 25471098*y^7 + 12576760*y^6 + 955878*y^5 - 232127*y^4 - 16898*y^3 + 1505*y^2 + 112*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 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*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]])