\\ Pari/GP code for working with number field 46.0.32506941609913674715356871213478525555718840947168228744163199689015235118467271106844101910141694545745849609375.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^46 - y^45 + 141*y^44 + 227*y^43 + 8537*y^42 + 36295*y^41 + 345604*y^40 + 2076564*y^39 + 11864192*y^38 + 69814704*y^37 + 342020403*y^36 + 1686823941*y^35 + 7622645751*y^34 + 32096474663*y^33 + 129794027533*y^32 + 486943520590*y^31 + 1733001170253*y^30 + 5832121065743*y^29 + 18427711979635*y^28 + 55182334783659*y^27 + 155781969455663*y^26 + 414534296781226*y^25 + 1042583697918161*y^24 + 2470416120266261*y^23 + 5519840948836859*y^22 + 11635807627174522*y^21 + 23104744569940586*y^20 + 43249996274928622*y^19 + 76336786345134775*y^18 + 126914799027058213*y^17 + 198792327266045961*y^16 + 293346142124093485*y^15 + 407056632927436565*y^14 + 530864837815463701*y^13 + 649874707667136778*y^12 + 742544430849406744*y^11 + 788328794242744596*y^10 + 767979325881817767*y^9 + 668328053991104215*y^8 + 502872891819354325*y^7 + 307890612187253725*y^6 + 132862824278047866*y^5 + 41589332567717579*y^4 + 48569951247045462*y^3 + 64040243988594722*y^2 + 28528684967581896*y + 4445208434181589, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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^46 - x^45 + 141*x^44 + 227*x^43 + 8537*x^42 + 36295*x^41 + 345604*x^40 + 2076564*x^39 + 11864192*x^38 + 69814704*x^37 + 342020403*x^36 + 1686823941*x^35 + 7622645751*x^34 + 32096474663*x^33 + 129794027533*x^32 + 486943520590*x^31 + 1733001170253*x^30 + 5832121065743*x^29 + 18427711979635*x^28 + 55182334783659*x^27 + 155781969455663*x^26 + 414534296781226*x^25 + 1042583697918161*x^24 + 2470416120266261*x^23 + 5519840948836859*x^22 + 11635807627174522*x^21 + 23104744569940586*x^20 + 43249996274928622*x^19 + 76336786345134775*x^18 + 126914799027058213*x^17 + 198792327266045961*x^16 + 293346142124093485*x^15 + 407056632927436565*x^14 + 530864837815463701*x^13 + 649874707667136778*x^12 + 742544430849406744*x^11 + 788328794242744596*x^10 + 767979325881817767*x^9 + 668328053991104215*x^8 + 502872891819354325*x^7 + 307890612187253725*x^6 + 132862824278047866*x^5 + 41589332567717579*x^4 + 48569951247045462*x^3 + 64040243988594722*x^2 + 28528684967581896*x + 4445208434181589, 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(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])