\\ Pari/GP code for working with number field 42.0.29198428620782310880522337720254845955751250559410488348634029682058779274295867292920491.1

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

\\ Define the number field: 
K = bnfinit(y^42 + 126*y^40 + 7371*y^38 + 265734*y^36 - 337*y^35 + 6608385*y^34 - 35385*y^33 + 120236886*y^32 - 1698480*y^31 + 1656597096*y^30 - 49362075*y^29 + 17647946044*y^28 - 969725925*y^27 + 147161484876*y^26 - 13614951987*y^25 + 966512559600*y^24 - 140844330900*y^23 + 5007866589747*y^22 - 1090864717741*y^21 + 20438068055202*y^20 - 6365649711708*y^19 + 65504728776627*y^18 - 27967555663866*y^17 + 164712579358698*y^16 - 92087076475296*y^15 + 327078003099919*y^14 - 226780519461930*y^13 + 522529550813886*y^12 - 423863032664799*y^11 + 688995831550950*y^10 - 628212651886323*y^9 + 777300365125620*y^8 - 764561060990476*y^7 + 834254335608561*y^6 - 771769032155877*y^5 + 864745657173342*y^4 - 917560666081710*y^3 + 525223978833708*y^2 - 936510480777564*y + 1669935825763633, 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 + 126*x^40 + 7371*x^38 + 265734*x^36 - 337*x^35 + 6608385*x^34 - 35385*x^33 + 120236886*x^32 - 1698480*x^31 + 1656597096*x^30 - 49362075*x^29 + 17647946044*x^28 - 969725925*x^27 + 147161484876*x^26 - 13614951987*x^25 + 966512559600*x^24 - 140844330900*x^23 + 5007866589747*x^22 - 1090864717741*x^21 + 20438068055202*x^20 - 6365649711708*x^19 + 65504728776627*x^18 - 27967555663866*x^17 + 164712579358698*x^16 - 92087076475296*x^15 + 327078003099919*x^14 - 226780519461930*x^13 + 522529550813886*x^12 - 423863032664799*x^11 + 688995831550950*x^10 - 628212651886323*x^9 + 777300365125620*x^8 - 764561060990476*x^7 + 834254335608561*x^6 - 771769032155877*x^5 + 864745657173342*x^4 - 917560666081710*x^3 + 525223978833708*x^2 - 936510480777564*x + 1669935825763633, 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]])